Constructing shadows in interior perspective. Constructing shadows in the interior

When constructing shadows in perspective drawings, the sun is taken as the light source, which can occupy different positions in relation to the picture:

1. the sun is located behind the object and the shadow falls towards the observer (Fig. 104);

Rice. 104. The sun is behind the object

2. the sun is located behind the viewer, the shadow falls towards the horizon line from the base of the object (Fig. 105);

Rice. 105. The sun is behind the viewer

3. the sun is located on the side so that the rays run parallel to the picture (Fig. 106).

Rice. 106. The sun is on the side of the object

The last case is most often used by engineers when constructing perspective images of buildings and structures, so we will dwell on it in more detail.

Let's consider constructing a point in perspective. We will assume that the object is illuminated from the left (or right), the rays go parallel to the picture, making an angle of 45° with the object plane. Let us write these conditions symbolically:

1. S ∥k;

2. S^ T= 45°.

Let's draw through the point A(Fig. 107) the perspective of the ray, and through its secondary projection (point a) – secondary projection of the beam. Since the ray is parallel to the painting, its secondary projection is parallel to the base of the painting t – t. The point of intersection of the ray's perspective with its secondary projection will determine the actual shadow of the point A on the ground - a point A T .

Rice. 107. Shadow of a point in perspective

Let's construct the own and falling shadows of the parallelepiped standing on the ground (Fig. 108).

Note that the conclusions that were formulated earlier for constructing shadows in orthogonal projections are also valid for central ones.

Rice. 108. Constructing shadows of a parallelepiped

Let's analyze the illumination of the faces of the parallelepiped. For a given direction of the ray flow, the top, left visible and invisible edges of the object in the drawing will be illuminated. The remaining edges will appear in their own shadow. Let's determine the contour of the body's own shadow. It will include the ribs [ 12 ] – [23 ] – [34 ] – [45 ] – [56 ] – [61 ], forming a closed chain in the form of a spatial broken line. From the identified contour we construct a falling shadow. Since point 1 lies on the ground 1 = 1 T. Let's draw through the point 2 perspective of the ray, and through its secondary projection (point 1 ) – its secondary projection. At the intersection of these lines we find a point 2 T. Since the edge [ 23 ] parallel to the object plane, its cast shadow is equal and parallel to it. Edge vanishing point [ 23 ] is on the horizon line (point F 1 ). Connecting the dot 2 T with this point (i.e. draw a line through it parallel to this edge). On the same line is the shadow of the point 3 . Let's draw through the point 3 perspective of the ray until it intersects with the constructed line - determine the point 3 T . In this case, a secondary projection of the ray should not be constructed, since the desired point has already been established by the intersection of two lines. Rib [ 34 ] also parallel to the plane T, its shadow is parallel to the edge.

The vanishing point of these lines is the focus F 1 . Drawing the perspective of a ray through a point 4 to the intersection with the segment [ 3 T F 1 ], define the point 4 T. Points 5 and 6 are located on the object plane T, That's why 5 = 5 T And 6 = 6 T. The outline of the contour of the falling shadow of a parallelepiped consists of a set of segments [ 1 T 2 T ] – [2 T 3 T ] – [3 T 4 T ] – [4 T 5 T ] – [5 T 6 T ] – [3 T 4 T], representing a closed loop.

Let's consider the tasks associated with constructing perspective and shadows of building fragments

Task 1

Construct shadows from straight barriers on the stairs, ground and wall (Fig. 109).

Rice. 109. Staircase with straight barriers

First, let's construct the shadows of the right barrier (Fig. 110). Since for a given direction of the light flux the right side of the barrier is in its own shadow, it is easy to see that the edges located on the border of light and shadow will be part of the contour of its own shadow. Let's define the falling shadow of a vertical edge. Dot A belongs T, That's why it may be noted that A = A T. Let's draw through the point IN the perspective of the ray, and through its secondary projection - the point A secondary beam projection perspective. At the intersection of the constructed lines we define a shadow IN T . Another rib [ B.C.] parallel to the object plane, therefore, its shadow is parallel to the edge and has the same vanishing point F 2 . The real part of this shadow on the ground is the segment [ IN T 1 T]. Since the point 1 T located on the border between the land and the wall 1 T = 1 T " . Using the back ray, you can determine a point on the edge [ B.C.], which cast this shadow. Dot WITH horizontal edge is on the wall, so WITH = WITH T " . Shadow of the segment [ 1 C] falls on the wall. Its shadow is the segment [ 1 T " WITH T " ].

Rice. 110. Constructing the contour of the falling shadow of the right barrier

The contour of one's own shadow is always closed. Reasoning based on its definition was given in many problems. An outline element can coincide with its shadow (if, for example, it is on the ground, a wall, or adjacent to another object). This factor should be taken into account when constructing a falling shadow.

At the left barrier, the right side is in its own shadow, therefore, the edges [ LN] And [ L.M.] are part of the defined contour (Fig. 111). Let's construct the falling shadows of these edges.

Rice. 111. Constructing the contour of the falling shadow of the left barrier

Radial plane (frontal plane of the level) passing through the edge [ LN] crosses the ground and the bottom step in parallel straight lines, leaving shadow marks on them, and the riser in a vertical straight line. Top point L This edge casts a shadow on the first step and is determined by the intersection of the ray with its secondary projection. Rib [ L.M.] parallel to the plane of the bottom step, so its shadow is parallel to the edge. Connects the dot L T with a vanishing point F 2 and mark the real part of the shadow of this edge on the bottom step to the point 2 T = 2 T " . Note that this edge is with a nail in relation to all risers. Let's carry out auxiliary lines to find common points for the edge [ L.M.] and the edges of all risers. These constructions will allow you to determine the falling shadows on the risers. In Fig. 111 on edge [ L.M.] all its sections are marked, casting shadows on specific fragments of the stairs, the ground and the wall.

Rice. 112. Own and falling shadows from direct barriers

In Fig. 112. The final version of the solution to the problem is presented.

Shadows of the ribs [ L.M.] And [ B.C.] on the wall and risers are parallel and represent an example ascending straight lines. Their vanishing point is located above the horizon line, and the vanishing point of their secondary projections lies on the horizon line.

Task 2

Construct a perspective of the roof eaves and determine its own and falling shadows (Fig. 113).

Rice. 113. Condition of task 2

Let us indicate the position of the picture plane on the orthogonal drawing of the problem conditions and select the point of view in accordance with the recommendations given earlier.

To solve the problem, we will use the architects’ method and use some other techniques for constructing perspective. Let's determine the starting points of the direct dominant directions and mark them on a perspective drawing based on the picture. Let us determine the vanishing points of these lines.

By connecting the starting points with the corresponding vanishing points, we obtain the perspective of a flat figure (roof eaves plan). Let's walk through the point of view and points 2 And 4 rays, which, together with their secondary projections, define horizontally projecting planes that intersect the picture along vertical lines (Fig. 114).

Rice. 114. Application of two methods of constructing perspective

In accordance with these considerations, the perspective drawing

let's draw through the points 2 1 And 4 1 vertical lines along which the constructed planes will intersect with the picture. An edge that falls into the picture plane will be depicted on it in natural size, taken from the orthogonal drawing. Drawing straight lines through the top and bottom points of this edge to the vanishing points F 1 And F 2 , let's complete the construction of the two visible lateral edges of the cornice (Fig. 115).

Rice. 115. Construction of the side faces of the cornice

using the method of conic sections

Let's draw two straight lines through the lower points of the vertical side ribs of the cornice to the vanishing points F 1 And F 2 , and select the outline of the lower edge (Fig. 116).

Rice. 116. Drawing straight lines perpendicular to the picture

To construct the perspective of the walls, straight lines are used, perpendicular to the picture, passing through the points 5 , 6 And 8 .

Rice. 117. Building visible walls in perspective

After finding the secondary projections of these points on the perspective drawing, draw vertical lines through them (Fig. 116).

Let's move one of the vertical edges into the picture plane in any direction. Let's put it on it from the base of the picture from the point 5 0 the actual size of the rib, taken from the orthogonal drawing (Fig. 117).

Let us draw a straight line through the top point of this edge to the vanishing point F 2 . Let's outline the right wall. Then we will construct parallel lines with a vanishing point F 1 and outline the left wall.

Rice. 118. The final stage of building perspective

In Fig. 118. the final result of constructing the perspective of the structure is shown.

Let's move on to building shadows. Let's determine the illumination of the object's edges for a given direction of the light flux and highlight its own shadows. Let's construct a falling shadow of the roof eaves on the walls. Let's find the shadow of a point A on the left visible wall. Let's draw through the point A ray perspective, and through A secondary projection to the intersection with the left wall. Note that the ray and the edge are intersecting lines. The intersection of the drawn ray with the wall will occur at the point A T " . Since the lower front edge of the left edge of the cornice is parallel to the left wall, its shadow will go along the wall to the right of the point A T " parallel to this edge. Therefore, through A T " and vanishing point F 1 we carry out a direct line.

At the point A three ribs of the cornice converge. His lower left rib is with a nail in relation to the left wall. Let's define the shadow of this edge. In Fig. 119 shows two options for finding a shadow.

In the first case (Fig. 119, A) on this edge we construct a point using the inverse ray IN which will cast a shadow IN T " on the left vertical edge. The shadow of the nail is the segment [ A T " IN T " ].

In the second case (Fig. 119, b) found a common point for the left wall nail. To do this, the upper horizontal edge of the left wall is extended until it intersects with with a nail and the point is marked WITH T " . Since the segment [ WITH T " A T " ] lies in the plane of the wall and intersects its left vertical edge, a point can be marked on it IN T " and highlight the real part of the shadow of the nail.

Both methods give the same result.

Rice. 119. Options for finding the falling shadow of the cornice

on the wall of a building:

A– using a point B T " ;

b– using a point WITH T " (“base” of a nail on the wall)

In Fig. 120 shows the perspective of this structure when choosing a different point of view, in which the shadow of the point A falls onto a wall invisible in the picture. In relation to this wall the edge [ AB] is with a nail and partially casts a shadow on it in the form of a segment [ WITH T " A T " ]. On the left wall there is a shadow of the lower edge of the visible left edge of the cornice.

The construction of the shadows of the cornice on the fragments of the structure was carried out in various options, because it causes difficulties for students when performing work.

Rice. 120. Constructing the shadow of the cornice with a changed point of view

Let's construct the falling shadow of the cornice on the ground separately from the lower part of the structure (Fig. 121), having previously determined its own shadow contour.

Rice. 121. Falling shadow of the cornice

Then we will find the contour of our own shadow and determine the contour of the falling shadow of the building without taking into account the cornice (Fig. 122).

Let's outline the general contour of the falling shadow of the structure and highlight it with color (Fig. 123).

Rice. 122. Contours of falling shadows of two objects

Rice. 123. Own and falling shadows of an object

The color of the falling shadow depends on the object on which it appears (on grass, asphalt, etc.) and has a thicker shade compared to its own shadow, as shown in the figure above.

Task 3

Based on the given views of the building, create a view on the left and construct own and falling shadows (Fig. 124).

Rice. 124. Condition for problem 3

Let us show on the building plan the position of the picture plane, the point of view, the vanishing points of parallel straight lines of two directions and draw auxiliary straight lines to construct a perspective (Fig. 125).

Rice. 125. Choosing a picture and point of view on the building plan

Rice. 126. Perspective of the visible walls of the building

Let's plot the starting points of the lines based on the picture. Let's construct a perspective of the visible walls of the building (Fig. 126).

Let's create a niche in the facade wall. Fragments of a niche with construction lines are shown in Fig. 127.

Rice. 127. Prospects for niche fragments

On an edge lying in the picture plane, we will draw division points to construct windows and connect them to the vanishing point F 1 . To build vertical lines, we use straight lines perpendicular to the picture, with a vanishing point P(Fig. 128).

Rice. 128. Formation of windows in perspective

Parallel straight lines with the vanishing point are drawn through the division points on the lower edge of the niche F 2 . On the rear edge of the niche, vertical straight lines are built and window compartments are outlined (Fig. 129).

Rice. 129. Fragment of window depiction

Using the lines drawn on the plan, we begin building the steps (Fig. 130).

Rice. 130. Start building steps

Using the actual dimensions of the vertical segments on the picture plane, we outline the profile of the steps and the right part of the canopy (Fig. 131).

Rice. 131. Construction of the profile of the steps and part of the canopy

We build the left part of the stairs and the canopy (Fig. 132).

Rice. 132. Construction of the left fragment of the building

In Fig. 133. shows an enlarged fragment of part of the visor, on which the edge located in its own shadow is visible,

Rice. 133. Left side of the visor

In the above drawings, the images showed their own shadows for a full perception of the drawing. No explanations were given regarding their constructions, since a sufficient number of problems on this topic were previously considered.

Rice. 134. Constructing a falling shadow of a canopy on the wall of a building

The falling shadows of the visor (Fig. 134) should be built from those edges that are on the border of light and shadow. This boundary (the contour of its own shadow) is clearly visible in Fig. 135.

Rice. 135. Fragment of a visor with its own and falling shadows

The elements of this contour are the lower front edge of the visor, parallel to the wall, and the lower left edge, perpendicular to the wall. Dot A is common to these edges. To find the shadow, we draw a ray through it and build its secondary projection. The intersection of the beam with the wall will occur at the point A T " . Draw a straight line through this point to the vanishing point F 1 . Using the back ray we determine the point IN on an edge perpendicular to the wall, which will cast a shadow on left rib walls. Line segment [ A T " IN T " ] – falling shadow nail on the wall.

In Fig. 136 it is clear that the edges of the staircase profile, parallel to the ground, and their shadows have a common vanishing point F 2 , edge [ 45 ] casts a partial shadow on the wall, starting from a point 6 , found using the backward beam.

Rice. 136. Falling shadows from steps on the ground and wall

To find the shadow of the visor in a niche, you can proceed as follows. First, construct a complete outline of the falling shadow on the wall without taking into account the niche (Fig. 137). Define the shadow of a point A on the plane of the wall (point A 1T " ). Connect the constructed point with IN T " and draw the real part of the shadow of the nail on the wall. By moving the point A 1T " deep into the niche until it coincides with its back edge, we will find the shadow of a point on it A(point A 1T " ).

It was possible to carry out the construction in reverse order. First determine the shadow of the point A in the niche of the window (point A T " ). Then find the shadows of the vertical and horizontal edges in it.

In Fig. 138 a shadow is visible on the window sill and on the window glass from the front vertical edge of the side edge of the niche.

Rice. 137. Falling shadow of the canopy on the wall and in the niche

Rice. 138. Fragment of constructing the falling shadow of the visor

On the right side of Fig. 138 it can be seen that the secondary projection of the ray passing through the point A, intersects the secondary projection of the rear edge of the niche. A vertical line is drawn through the intersection point, on which a point is marked A T " .

Rice. 139. Constructing a falling shadow of a building on the ground

When determining the falling shadow of a building (Fig. 139), the edges included in the contour of its own shadow are used. This is a vertical edge located in the picture plane, the top right visible edge with a vanishing point F 2 and the top invisible edge with the vanishing point F 1 . The shadows of these edges on the ground are parallel to the edges themselves and have the same vanishing points.

Rice. 140. Perspective of a building with its own and falling shadows

The completed image (Fig. 140) shows that the falling shadows acquire the color of the surface on which they are cast, but the color tone becomes denser.

WITH CENTRAL LIGHTING (Fig. 18)

Construct the interior and shadows in perspective according to Fig. 18 (the task is common to all students).

Explanations:

The light source is usually designated by point S*, in our case it is a pendant lamp that is attached to the ceiling at point S*. The construction of shadows is best considered using the example of a single point.

Construct a falling shadow from a certain point A of space onto the object plane H. We enclose parallel segments S*S and Aa in an auxiliary plane R and in this plane draw a light ray from the light source S* through point A until it intersects with the object trace Rh at point A* ≡M. The object trace of this ray at point A* will be the falling shadow from point A onto the object plane H. Consequently, the falling shadow of point A onto the object plane H is the object trace of the light ray emanating from the light source S* and passing through given point And space.

The construction of a falling shadow is no fundamentally different from a rectangular plate. The solution is presented graphically in Fig. 12.

In the proposed interior there is a case of constructing a falling shadow from the inclined plane of the picture onto a vertical wall. Then the construction is carried out in the sequence as in Fig. 12. The light plane S* AB should be drawn through the point S* and the segment AB; obviously, the shadow from the plane will be its continuation. The object trace AB=M1M2 of the shadow plane AA*B is the falling shadow of the segment AB on the object plane.

Directions for work:

1. construct a perspective of the interior and the objects located on it, the dimensions of which are taken arbitrarily, but taking into account the composition of the sheet.

2. When constructing a falling shadow from a hanging picture, conditionally take the plane of the vertical wall as the object plane and, accordingly, Fig. 12, make constructions.

3. Determine your own shadows of objects.

4. Construct falling shadows on the horizontal plane of the floor and vertical planes of the walls.

5. Make the interior color scheme yourself in compliance with the laws aerial perspective and color science.

EPUR 2

Topic: CONSTRUCTION OF PERSPECTIVE OF SHADOWS FROM ARCHITECTURAL

OBJECTS IN SUN LIGHT (Fig. 19)

Construct, using the architects' method, the perspective of a building and shadows in sunlight parallel to the picture. Take data from table No. 5 according to your option.

Directions for work:

1. Based on these dimensions, construct two types of object - facade and plan.

2. In the orthogonal drawing, determine the elements of the perspective apparatus: set the point of view S, the main perpendicular of the picture SP and the bases of the picture perpendicular to it kk. Using the vertical and horizontal lines, we determine the vanishing points F1 and F2 of the horizon line hh.

4. Transfer the given elements to the picture and, using the architects’ method, construct a perspective of the building.

5. Identify and construct the building's own and cast shadows.

6. When cleaning the building, make the falling shadows darker than your own.

EPUR 3

Topic: PERSPECTIVE OF REFLECTIONS IN FLAT MIRROR SURFACES (Fig. 20)

The diagram consists of 2 tasks.

1. Based on the data in Table No. 6, construct an angular perspective of the interior and its reflection in a flat vertical mirror.

2. Based on the data in Table No. 7, construct the perspective of reflections in a calm water surface (a flat horizontal mirror).

Explanations:

You need to remember the basic physical laws of light reflection from flat mirror surfaces:

1. The incident beam SK and the reflected beam KE lie in the same plane with the normal AK drawn perpendicular to the reflecting surface of the mirror BB (Fig. 13).

2. Angle of incidence equal to angle reflections α=β.

Figure 13c shows the reflected rays of light AB and A1b. A viewer looking into a mirror perceives with his eye the reflected rays Аb and А1b and will see in the mirror BB point S" at the intersection of the reflected rays at point S0, which is called the mirror reflection of point S.

Figure 13 shows that points S and S" are on the same perpendicular to the reflecting plane and are located on equal distances from the base of the perpendicular to point S i.e. Ss=S"s. The construction of an image in a flat mirror is based on this.

Directions for work:

1. Two tasks are performed on one sheet.

2. Place the format vertically. Split it horizontally thin line taking into account the stamp on the field of the lower drawing.

3. When constructing a reflection in a flat vertical mirror, move the interior itself slightly to the left to preserve the construction lines in the drawing.

4. When constructing reflections in a flat water surface, place the vanishing points F1 and F2 of the main directions of the objects on the drawing field.

5. Cleaning should be carried out taking into account the laws of painting.

An artificial light source, like any point in perspective, is defined in the picture as the perspective of the luminous point itself and the perspective of the base ( see fig. 9.22).

The light source can be located anywhere relative to the illuminated object. It depends on how the artist wishes to use light in the composition of the painting.

The length of the shadow depends on the height of the luminous point and its distance to the illuminated object. The shadow should not extend beyond the horizon line or ABOUT-ABOUT. If it is above the horizon, it is an imaginary shadow. Therefore, you need to choose the right light source.

If an object is illuminated by several light sources, then the falling shadows overlap one another. The place where two falling shadows overlap is called full shadow . The mismatched parts of the falling shadows are called penumbra . First they build their own shadow, then the penumbra, then the full shadow, but not the black one, since it is illuminated by reflected light.

Example 1. Construct a falling shadow from the vertical for two given light sources ( rice. 9.27).

Solution

1. Determine the boundary of your own shadow. For a given position of the light sources, the edges of the shadow will be the boundary V" K V K And E" K E K, i.e. in its own shadow there will be edges A" K A K B" K B K And A" K A K E" K E K.

2. Construct falling shadows from edges A" K A K B" K B K And A" K A K E" K E K first from the first light source, and then from the second.

3. Determine the boundary of the full shadow and penumbra.

Example 3. Construct your own shadow and a falling shadow from a vertical cylinder. The position of the light source is determined by the perspective and the perspective of the base ( rice. 9.29).

Solution

1. Determine the zone of your own shadow. From point C" K(perspective of the base of the source) draw tangents to the lower base of the cylinder. Generators of the cylinder drawn from the points of tangency 1 TO And 6 K, will limit the area of ​​their own shadow.

2. Let's build a falling shadow. To do this, we divide the arc of the base of the cylinder in the unlit part into an arbitrary number of sections of arbitrary length with dots 2" K, 3" K etc.

3. Let's draw generators through these points and construct shadows from these generators. Line 1 T-2 T-3 T-4 T-5 T-6 T will limit the area of ​​the falling shadow.

Constructing shadows in the interior

When depicting interiors, artificial lighting is most often used. Solar lighting in the interior is used only if there are large light openings (terraces). If the windows are of normal sizes, then the light “bunny” can be neglected.

Rule for constructing shadows

To find a shadow from a point, you need to draw a ray through the light source and the point and find the point of intersection of this ray with the plane on which the shadow falls. To do this, solve the problem of the intersection of a line with a plane. We draw an auxiliary projection plane through the light beam: if the shadow is on the floor, then the plane is horizontally projecting; if on vertical walls, it is frontally projecting.

Example 1. Construct a shadow from vertical lines on the floor and side wall of the room at a given position of the luminous point ( rice. 9.30).

Solution. In this example, it is convenient to draw horizontally projecting ray planes. The horizontal trace of these planes will pass through the perspective of the base of the light source and the perspective of the base of the points A And IN. The point of intersection of the trace of the plane with the light ray gives the shadow of the point A on the floor. This construction is called the sail method.

9.3.4. Constructing shadows from objects on various surfaces
in natural and artificial light

Example 1. Construct a falling shadow from the balcony on a vertical wall in natural light ( rice. 9.32).

Solution

1. Determine the zone of your own shadow. With a given light source, the right side wall of the balcony and the lower part of the floor will be in their own shadow.

2. Construct falling shadows from the contour of our own shadows. To do this from the points BK, G K And LK Let's draw light rays at an angle of 45° and determine the points of intersection of these rays with the vertical wall of the house.

To determine the points of intersection of light rays with a vertical wall, we determine the perspectives of the base of all points of the balcony on the object plane (points A" K, M" K, L" K, E" K, J" K, B" K, G" K).

Through the perspectives of the base of the points B" K, G" K, L" K Let's draw the perspective of the base of the light rays until they intersect with the vertical wall (point 1 And 2 ). From points 1 And 2 Let's restore the perpendiculars until they intersect with the light rays drawn from the points B" K, G" K, L" K. Let's connect the obtained points B" K, G" K, L" K. These will be the shadows from the ribs B K G K, G K L K. Connecting V T With E K, we get the shadow from the edge L K M K.

Example 2. Construct a vertical drop shadow AB to the object plane N and onto the surface of a truncated prism ( rice. 9.33).

Solution. Since the point IN vertical belongs to the object plane, the shadow of the point IN coincides with the point itself IN. Thus, solving the problem comes down to constructing a shadow from a point A.

1. Through the perspective of a point A (A K) and source perspective ( S K) hold the perspective of the light beam. Dot ( A T) – hypothetical location of the shadow from the point A on the object plane, if there were no obstacle in the path of the light rays.

2. Through the perspective of the base of the point A (A" K) and the perspective of the base of the source ( C" K) draw a perspective of the base of the light beam.

3. Construct a line of intersection of the horizontally projecting plane of light rays (plane CAB passing through the vertical AB and light source WITH) with the surface of a truncated prism – line 1 K 1" K 2" K 2 K.

4. Vertical shadow AB will go from the shadow of the point IN onto the object plane (coinciding with the point itself IN), along the perspective of the base of the light beam until it intersects with the surface of the prism (point 1 TO). Next - along the line of intersection of the plane of light rays with the surface of the prism. The boundary point of the shadow ( A T) will be the point of intersection of the line 1 K 1" K 2" K 2 K with light beam perspective.

Bibliography

1. Makarova, M. N. Perspective / M. N. Makarova. – M.: Academic project, 2006.

2. Ivashina, G. G. Perspective / G. G. Ivashina. – St. Petersburg: SPbGHPA, 2005.

3. Solovyov, S. A. Drawing and perspective / S. A. Solovyov. – M.: graduate School, 1967.

4. Kotrubenko, M. E. Collection of problems for the course “ descriptive geometry And technical drawing» / M. E. Kotrubenko, O. K. Leskova, L. N. Karagezyan. – St. Petersburg: IPC SPGUTD, 2006.

Related information.

Lecture 24 Constructing shadows in the interior Position of the light source Constructing shadows of geometric bodies Inverse ray method Ray section method

The construction of shadows in the interior is quite difficult task. This is explained, firstly, by the presence of various lighting sources - solar, diffused and artificial light and, secondly, under conditions of illumination with artificial light sources, a large number of them, a variety of shapes and locations in modern interior make the task of accurately constructing shadow contours quite difficult.

Three cases of constructing shadow contours Depending on the type of interior lighting sources, three cases of constructing shadow contours are possible: With sunlight penetrating through window openings; With point light sources; In diffuse daylight

Constructing shadows in sunlight Task 4. 2 p. 34: Construct a sunspot from the contour of a rectangular window opening (the thickness of the walls is specified and taken into account during construction) The sun is in front of the viewer

Sequence of construction: 1. Construct a falling shadow from the inner contour of the opening: from vertical edges 1 and 2, shadows fall along the projection of the beam, from horizontal edges 2 -1 - in parallel. 2°

2. We build a falling shadow from the external opening (from vertical edges 4 and 3 - along the projection of the beam; from horizontal edges 4 -3 in parallel. We get overlays of shadow points 5 o and 6 o The shadow from edge 4 -3 (4 o-3 o) is superimposed on the shadow from edge 1 -1 at point 6 o. 2° ° °

3. Using a reverse beam, return point 5 o to the horizontal edge 2 -1 of the window sill. Return (.)6 o to the vertical edge 1 -1 ° ° 2° ° °

4. Edge 4 -3 rests on the right side wall at point 3 - the shadow closes. The shadow on the window sill from edge 4 -4 falls in the direction of the secondary projection of the beam. ° ° 2° Sunny “bunny” ° °

Creating shadows in sunlight sunlight, penetrating through a rectangular window opening, forms a clear and contrasting quadrangle on the floor.

Constructing shadows with a point light source With a point light source, the ray lines are not parallel to each other and do not have vanishing points, they intersect at the “luminous” point of the light source Falling shadows are constructed using the secondary projection of the light beam

Problem 4. 4 p. 36: A vertical plane is given in the picture. It is required to construct a shadow from a plate with a point light source

If we take another light source - S*, then an overlay of falling shadows will occur. S* ° Во ° ° S 1* ° Ао

The final drop shadow is determined by general outline. The shadow at the place of the overlay will be darker S* ° Во ° ° S 1* ° Ао

Problem 4. 5 p. 36: The picture shows a vertical plate and a rod resting on its upper edge. It is required to construct a shadow from a plate and a rod with a point light source

Solution: 1. Let's construct a shadow from an inclined line: Let's draw a light ray through (.)S' and (.)A', and the secondary projection of the ray S' 1 and A' 1 and find their intersection. Ao‘

Since the straight line AC rests on the plane of the floor, the shadow at the point of support in it itself is C'= C 1'= Co' By connecting the points Co' and Ao' we get a shadow from the straight line to the floor

2. At point B, the rod rests on the plate - the shadow closes 3. Construct the shadow of the plate

Problem 4. 6 p. 37: The picture shows the perspective of a prism and a rod resting on its upper edge. It is required to construct a shadow from a prism and a rod with a point light source

2. Determine your own shadows on the prism. Constructing a falling shadow from a prism 2 1 21 11 1 o 2 o

3. To determine the shadow from the inclined straight line AB onto the upper plane of the prism, you can use: a) the reverse ray method: we return the point of overlap of the shadows from the straight AB to the shadow from edge 2 -3 (Mo) to edge 2 -3 3 m mo 1 11 2 21 1 o 2 o

Problem 4. 7 p. 37: in the picture it is given triangular prism and a straight circular cone. It is required to construct a shadow from them with a point light source

Solution: 1) To construct the shadow of a cone, find the shadow of its vertex (.)T‘ -To‘

2) Determine the falling shadow: draw tangents from (.)To‘ to the base of the cone, then determine our own shadow. 3) Using the ray section method, we determine the shadow from the top of the cone onto the inclined plane of the roof

The second option for constructing a shadow from a cone onto a prism: using the inverse ray method (we return points 1 o and 2 o of the shadow overlay from edge B and the cone to edge B’) °° ° °

When constructing shadows in the interior perspective, you should first construct projections of the light source onto those enclosing planes of the interior on which you will need to construct shadows: floor, ceiling, walls

Task 4. 8. p. 38: Construct projections of a point light source onto the vertical planes of the walls and floor in a given frontal perspective of the interior

Solution: 1) We determine the projections of the light bulb S onto the walls, floor and ceiling (through the light source we draw perpendiculars from (.)S to these planes. Since the frontal perspective of the interior is a plane perpendicular to the side walls, floor and ceiling, parallel to the picture) .

Example: Light source L. Vertical straight line Вв is perpendicular to the floor, therefore the shadow falls along the projection of the beam on the floor to the wall and vertically along the wall. °

L 1“ – projection of the light bulb onto the left side wall. With its help, we construct a shadow from the straight line “A.” °

L‘ - projection onto the end wall - since the side walls are perpendicular to the end wall, the shadow from horizontal straight openings falls along the projection of the ray onto the end wall drawn through L‘ Point of contact in the end plane ° ° Point of contact in the end plane

Task 4. 9 p. 38 b): Construct shadows from furniture with a point light source on the frontal perspective of the interior

From the vertical line 1 -11 the shadow falls along the projection of the ray, from the horizontal edge of the step - parallel and closes to the stop point. Point of stop

We determine the projections of the luminous point S on the plane of the steps (S 2, S 3, S 4). To do this, draw a plane parallel to the picture through the light source and determine the height of the steps at a given depth

We determine the lighting of the steps and build our own shadows. The vertical plane of the third stage is located in the same plane with point S (sliding beam). The vertical plane of the fourth stage is illuminated. Using (.) S 2 we build a falling shadow from the vertical edge 2 -21

From straight N-M on the rear end wall the shadow is parallel, then closes at the stop point M≡Mo. We construct a falling shadow from the cabinet using its secondary projection on the floor. Find the shadow from edge 1 -2 (1 o-2 o)

Edge 1 -3 is parallel to the wall, therefore its shadow falls parallel to the wall, i.e. we build using (.)P 4

Horizontal edge 2 -4 is also parallel to the plane of the wall. We build a shadow 2 o-4 o using point P. Next, the shadow closes at the point of contact of the straight line 4 -5 into the wall. Stop point

To construct a shadow from a vertical line A, we determine the projection of the light source onto the podium (Sp) using an arbitrary vertical plane (point F is taken arbitrarily)

The shadow from the straight line on the podium falls in the direction of the beam projection, on the vertical wall - parallel to the straight line

Task 4. 9 p. 39 c): Construct shadows from furniture with a point light source on the frontal perspective of the interior

Determine the shadows from points A and B (Ao 1 on the floor, Bo 2 on the wall)

We determine the break by constructing the shadow from (.)L and the closure of the shadow on the right wall C=Co Point of emphasis

We determine the falling shadows from the columns on the wall and on the ceiling (closed at the point S≡Sp); to construct a shadow on the balcony, we find the projection of the light bulb to the floor level of the balcony Sb ≡Sп ° Sb

To construct the falling shadow from the balcony onto the columns, draw an imaginary tangent plane to the columns and determine the lines of tangency on the columns. Imaginary plane tangent to the columns

Draw a shadow from a horizontal edge passing through (.)A on the imaginary plane using (.)P

At the intersection of this shadow from edge “A” with the tangents on the columns, we fix the points of the actually existing shadow (peak points)

We find the overlay of shadows from the columns and the balcony - points 1 o and 2 o and using the inverse ray method we return them to the contour of the columns’ own shadow - points 1 and 2 ° 2 1 ° ° 1 ° ° 2 o

Task 4. 10 p. 40: Construct projections of the light source onto two vertical planes of the walls, floor and ceiling in the angular perspective of the interior

Angular perspective of the interior. Method of combining the object plane with a picture Solution: Let's consider the first option - the room has a 90° angle in plan. C is the light source on the floor plan. Let us draw straight lines parallel to the walls of the room through (.)C and determine (.)1 and 2 picture traces of these straight lines 1 2

Constructing projections of a light source in a corner interior We construct perspective projections of a light source C using straight lines parallel to the sides of the plan: We construct the perspectives of these straight lines The intersection of perspectives of straight lines gives (.)Sp - projection (.)C on the floor we determine the nearest points 1 and 2 in the picture on ceiling

Constructing projections of a light source in a corner interior Constructing straight line perspectives The intersection of straight line perspectives gives (.)Sp - projection (.)C on the ceiling At an arbitrary distance we “hang” the light source C Sp ° ° C

Constructing projections of a light source in a corner interior To construct a projection (.)C onto wall P 2, you need to draw a perpendicular to it. Since the angle between the walls in plan = 90°, the perspective of a straight line perpendicular to the wall is constructed using (.) F 1 we determine (.) C 2

Constructing projections of a light source in a corner interior We similarly determine the projection of a light bulb on the right side wall C 3 (using (.) F 2.) ° C 3

Var. 2: Constructing projections of a light source if the angle between the walls on the floor plan is α≠ 90°. Perspective projection (.) C can be constructed using straight lines parallel to the walls of the room, i.e. using vanishing points F 1 and F 2 To determine projections draw the light source through (.)C straight lines m and n, perpendicular to the walls of the room

Construction of projections of a light source at an angle between the walls α≠ 90° on the floor plan. Let us determine the vanishing points of straight lines m and n, for which, through the combined point of view with the picture (.)S', we will draw straight lines parallel to m and n and find their intersection with the horizon line ( Fm and Fn respectively)

Construction of projections of a light source at an angle between the walls α≠ 90° on the floor plan. Using the vanishing point Fm, we find the projection C of 2 points C on the side plane

Construction of projections of a light source at an angle between the walls α≠ 90° on the floor plan. Similarly, we determine the projection C 3 of point C on the right side plane using point Fn

Construction of projections of a light source at an angle between the walls α≠ 90° on the floor plan. planes were constructed passing through the light source (.)C and perpendicular to the side walls to determine the projections of the lamp onto the side walls

Task 4. 11 p. 41: Construct shadows from a point light source in a given angular perspective of the interior

Solution: 1. The internal partition in the closet is in its own Shadow. We build a falling shadow from it using a projection on the floor

We determine the shadows from points 1, 2, 3. From (.)1 hit the wall, from (.)2 and 3 to the shelves
Constructing shadows with diffused lighting With diffuse, diffused light penetrating through a window opening, light is emitted over the entire area of ​​the opening. The contours of the shadows seem to overlap one another, their boundaries becoming more and more “blurred” as they move away from the light opening. The planes of the slopes are illuminated, therefore the vertical and horizontal edges of the slopes of the opening, facing the inside of the room, are shadow-forming.

Constructing shadows in diffused lighting From the many “luminous” points in the opening, points located in the corners of the opening (1, 2, 4, 5) are distinguished. Using points 1, 2 and 3, cast shadows on the floor, and using points 4 and 5 - on the ceiling. To construct shadows, it is necessary to project these points onto those planes of the room on which the shadows should be constructed: on the floor (points 1, 2), on the ceiling (points 4 and 5) and on the side wall (5"). Then draw from the “luminous” perspective points of ray lines through the shadow-forming points of the object until they intersect with the secondary projections of these rays

Constructing shadows with diffused lighting For example, let’s take “luminous” point 1, located in the upper corner of the opening. To construct a shadow from (.)A, it is necessary to draw a light ray through it and find its intersection with the projection of the ray on the floor. 1°° 11

Then we build shadows from AB and from BC ° 1 ° ° 11 Co ° Ao Vo

Let's take “luminous” point 2, located in the upper left corner of the opening. Let's construct shadows from points C and D and determine the shadow from straight line CD on the right wall. Let's complete the construction of the shadow from BC 2 ° Point of emphasis ° Co ° ° Ao Vo

Edge G of the inner part of the opening partially blocks the flow of light. Let's find the “luminous” point 3, located on the upper edge of the opening. To do this, we connect the projection of the vertical edge Ж (Ж 1) with the projection (.)А and extend it until it intersects with the projection of the outer side of the opening - (.)3¯ Ж ° С ° Ж 1 ° Ао Вo

Let's construct shadows from the vertical edge of the table leg E using the “luminous” point 3. We complete the construction of the shadow from the horizontal edge of the table passing through point E ° Point of emphasis in the lateral plane ° ° ° Ao Vo Co

Let's construct shadows from the horizontal edge of the LG opening using the “luminous” point 5 on the ceiling. g g ° Point of emphasis in the lateral plane of the wall ° ° С ° Ао Вo

Let's construct a shadow from the vertical edge GG 4 of the opening using the “luminous” point 4. On the ceiling the shadow falls along the projection of the beam, on the wall parallel to edge G). 44 ° G 4 f g ° Point of emphasis in the lateral plane of the wall 4 ° Co ° Ao Vo

Let's construct a shadow from the horizontal edge of the opening using the “luminous” point 1. The shadow falls on the floor parallel to the edge). f g ° ° ° Co ° ° ° Ao Bo ° °

Constructing perspective and shadows in perspective

Plan

1. Perspective of geometric bodies.

2. Choosing a point of view when constructing a perspective image.

3. Constructing a perspective image of the building.

4. Shadows in perspective..

1. PERSPECTIVE OF GEOMETRIC BODIES

Construction of a perspective image of a cube (Fig. 99). We draw the picture plane through the edge of the cube VM, in this case it will be projected on the picture plane in natural size. Let's set the position of the horizon line and make all the constructions similarly to the previous ones (Fig. 99). Vanishing points of straight lines AB,CD, AD And NE determined by the previously discussed method.

The transfer of points from the base of the picture plane to the picture is carried out as in the previous examples.

In a picture from a point V-M we restore the perpendicular on which we plot the natural length of the edge of the cube VM. We connect the extreme points of the edge with the vanishing points F 1 And F 2 , and from the points A To = E k and C k = G K we restore the perpendicular to the intersection with the lines representing the full perspectives of the lines coming from the edge VM to the vanishing points. Thus, we obtain a perspective image of the ribs AE And C.G.. To get an image of an edge DK, it is necessary from the extreme edges of the points AE And C.G. draw straight lines to vanishing points F 1 And F 2 . At the intersection of these lines we get edge points DK.

If the second vanishing point lies outside the drawing, for example the point F 2 , then you can build a perspective with one vanishing point F 1. To do this, we continue the horizontal projection D l A l until it intersects with the picture plane at the point N 1 , Full stop N 1 Let’s transfer it to the picture and from it we will construct a perpendicular, on which we will plot the natural height of the cube. Connecting the resulting points with the right vanishing point F 2 , we get a perspective image of the edges of the cube AE And DK as a result of the intersection of lines N l F 2 with perpendiculars AE And DK, reconstructed from the picture plane.

You can also construct an image of a cube if you use straight lines perpendicular to the picture plane, drawn through the vertices of the cube. In Fig. 99, b shows the construction of the perspective of two edges AE And C.G.. In this case, the main line of sight is directed like this. so that it does not coincide with the edge KD.

A perspective image can be constructed with a magnification of several times. for example, 2 or 4, etc. To do this, all dimensions, both vertical and horizontal, are increased when all points are transferred to the picture. Figure 100 gives an example of constructing a perspective image of two geometric bodies, a cube and a parallelepiped, located on the same level. The picture plane is drawn like this. so that two edges (one at the cube, the other at the parallelepiped) are projected on the picture plane without distortion, i.e. the picture plane is drawn through the edge 4 parallelepiped and edge A Cuba. The horizon line is drawn so that the top base of a cube is visible, while the top base of a parallelepiped is invisible.

We position the viewer so that the main line of sight is perpendicular to the picture plane (picture) and main point R was in the middle third of the picture.

Through all points of the figure we draw rays to the point of view and find the left and right vanishing points. Then we transfer the trace of the picture plane, together with all the points, to the place where the perspective image will be constructed.

In the picture, first we find the natural ribs 4 And A and from them we draw lines to the vanishing points. Drawing from the points 1 To , 2 TO , 3 To , D K , WITH To And IN To vertical straight lines, we find a perspective image of each point. By connecting them together, we obtain a perspective image of the given volumes.

2. CHOOSING A POINT OF VIEW WHEN CONSTRUCTING A PERSPECTIVE IMAGE

In order for an image to look good in perspective, it is necessary to take into account the person’s natural angle of view, so the relative position of the object, picture and point of view cannot be arbitrary.

When choosing a point of view, it is recommended to adhere to the following provisions:

Main beam The view should be directed perpendicular to the picture plane and divide the picture approximately in half or be in the middle third of the picture. That's what's called a painting. what will be contained between the extreme rays coming from the viewer to the object;

It is advisable to maintain the ratio AB/BC =A k B k / B k C k (Fig. 101);

U the gap between the base of the picture and the structure should be 20°...40°;

The viewer must be at such a distance from the object that the object is included in the cone of clear vision or is in the field of clear vision. To do this, the angle between the extreme rays of vision must be within 28°...37° (Fig. 102);

In the case when the vertical dimensions of a structure are larger than the horizontal ones, the viewer should move one and a half to two heights away from the structure so that the angle of view in the vertical plane is within the permissible limits (Fig. 103);

According to the location of the picture plane Regarding the object, perspectives can be of two types: central frontal perspective used for constructing interiors, i.e., perspective of the internal view of premises (Fig. 104); angular perspective(Fig. 105) is used when depicting individual objects, in this case the picture plane is located at an angle to the object.

According to the location of the horizon line perspective images can be (see Fig. 105, A): with normal horizon height, i.e. at a height of human height of 1.5... 1.7 m, it is used when constructing perspective on level ground (Fig. 105, b); when viewed from below used for individual parts observed from below, and for buildings standing on a hill (Fig. 105, V): with a high horizon, in this case, the horizon height is taken to 100 m and above (Fig. 105, G).

Based on the distance of the point of view from the subject, perspectives can be divided into perspectives with a sharp, sharp angle and perspectives with a blunt, flat angle. Foreshortening is the position of the depicted object relative to the picture plane, which results in a sharp shortening of parts distant from the foreground. The measure of perspective is the ratio of the perspective image of the ribs BB 0 in the foreground (see Fig. 106, A And b) to the edge A 1 A 0 the most distant edge of the same face BB 0 /A"A 0 .

When choosing a point of view, a necessary condition is the actual location of the point of view, i.e. the best. When choosing a point of view, you can use the following scheme (Fig. 107). When marking the standing points, mentally imagine what the building will look like. For example, dot 1 (see Fig. 106, 107) shows a side view of the building. The main part of the facade is hidden, point 2 reveals the main facade well, but is not visible sides; dot 3 gives a view of both facades, then since the perspective angle for both facades is the same, the perspective of the building turned out to be inexpressive; point 4 can be considered the most successful, since from this point of view the composition of the building is revealed in the best possible way.

3. BUILDING A PROSPECTIVE

BUILDING IMAGES

The perspective of any building (structure) consists of the perspective of many points, each of which is constructed as a trace of a ray of vision on the picture plane. There are several ways to build perspectives. The main ways to build perspective include:

1. a method of architects based on the use of vanishing points of parallel lines;

2. method of rectangular coordinates and perspective grid;

3. radial method and combined height method.

Each of these methods of constructing perspective uses different elements of central projection. The choice of one or another construction method depends on the type of object and its volumetric-spatial structure.

The architects' method is based on the use of vanishing points of perspectives of horizontal parallel straight objects and is used in practice to construct architectural perspectives.

The essence of the radial method of constructing perspective is to determine the points of intersection of the projecting rays with the picture plane. This method is used mainly in constructing frontal perspectives of streets, courtyards, and building facades with parts protruding forward.

The essence of the coordinate method is to construct the perspective of an object related to a rectangular coordinate system. The coordinate method is used when depicting simple objects of irregular shape.

The perspective grid method, as a type of coordinate method, is used in constructing “planning” perspectives with a high horizon when designing urban and industrial facilities located over a large area.

We will look at one of them - the architect's method. This method comes down to determining the projections of the points of the structure onto the picture plane by rays coming from the points of view to each point of the structure.

When constructing a perspective using the architect's method, the picture plane is placed at an angle to the building and its trace is drawn through one of the corners (Fig. 109).

The viewer is positioned so that the main line of sight is perpendicular to the picture plane, and the viewer himself is at such a distance that the angle of view , determined by the extreme rays of view S { and S 5 was equal to 23°...37". Main line of sight SP should divide the picture approximately in half so that the point R was in the middle third of the picture.

T vanishing points for the main directions of the plan can be found if we draw straight lines from the standing point S 1 parallel to the sides of the structure to re sections with the picture plane at points F 1 and F 2 .

Vanishing point F 1 (left) will be the vanishing point for all lines parallel to the sides 1-2, 3-4. 5-6, 8-9, and the vanishing point F 2 (right) – for parallel sides 1-7, 11-10, 2-3, 4-5 and parallel ones.

After installing the viewer, the picture plane and finding the vanishing points, rays of sight are drawn from all points of the structure and on the trace of the picture plane QC all intersection points are recorded 1 k... 6 K, etc.

To construct the perspective itself, we transfer the trace of the picture plane with all the points marked on it to the place where the perspective will be built (Fig. 110).

We draw the horizon line parallel to the base of the picture plane QC at a given height and transfer the vanishing points from the base of the picture plane to it.

Since the picture plane is drawn through the edge 4, then in the future it will be in natural length. From point 4 To we restore the nonpendicular to the trace of the picture plane and plot the height of the edge on it 4, taken from the frontal projection of the orthogonal drawing.

The lower and upper points of the rib 4 connect to vanishing points F 1 and F 2 . getting the direction of the sides of the building. Restoring perpendiculars from points 3k and 5 To before intersecting with the rays going to the vanishing points, we get the sides of the building. In the same way we find all the edges and sides of the structure in perspective.

To get points 8, 9, 10 to 11 in in the future we will continue the lines of the ridge 11-10 (see Fig. 109) until it intersects with the picture plane K K at the point N 1  , a line 8-9 to the intersection at the point N and move these points into perspective. From the obtained points we construct perpendiculars, on which we plot the heights from the ground to the ridge.

Connecting the dots N 1 And N 2 with vanishing points and intersecting the resulting lines with perpendicular straight lines constructed from the points 11 To , 10 To 8 To And 9 TO , we get a perspective image of straight lines 11-10 And 8-9, belonging to the roof ridges. We connect the found points, according to the orthogonal drawing, with the corresponding points, obtaining a perspective image of the roof.

So that the structure does not seem to be hanging in the air, it is necessary to draw a sidewalk, road, etc. near it, while ensuring that everything the drawn lines were directed to the vanishing points.

4. SHADOWS IN PERSPECTIVE

T Just like in axonometry, shadows in perspective can be constructed from different points of the light source.

In Fig. 111 shows eight possible arrangements of light sources relative to the position of the point of view and two vertical rods from which a shadow falls on the horizontal plane. Here the shadows are from the tops of the rods, i.e. from the points A And IN, found as horizontal traces of light rays passing through these points. From the examples considered, it is clear that shadows from vertical lines fall in the direction of the vanishing point on the horizon, and the length of the shadow is determined by the intersection of the ray of light passing through the upper end of the straight line to the vanishing point of the rays with the surface on which the shadow falls.

The direction of the light rays can be chosen depending on the nature of the object being depicted and the desire to show it illuminated from one side or the other. In this case, one should be guided by aesthetic considerations, since the construction of shadows on a project is not an end in itself, but only a means for identifying shapes and proportions.

In cases where the structure consists of arches and colonnades, it is good to use the so-called coming shadows. In this case, rays of light penetrating through the openings create a spectacular play of chiaroscuro.

Now let's determine the distance d, to which the vanishing point of light rays in space F 4 will be removed in the picture from the vanishing point of horizontal projections of rays F 3 . To do this, assume that the sun is located behind and to the left of the viewer, and the rays are directed down to the right, making an angle a = 35; 54". (At point S construct angle a and find the leg d right triangle SF 3 F 4, which is the desired value, and it should be plotted in the picture vertically down from the point F 3 of the horizon. All other constructions for finding shadows are clear from the drawing. To construct a shadow from a building that has a protrusion, we can recommend the following technique for choosing the direction of the light rays. Let's consider the construction (Fig. 112). To the corner 4 Apply a ruler to the ledge of the building KN so that the shadow falling from the ledge onto the facade 5-6 was either slightly smaller or slightly larger than the prospective projection size 4-5. and, drawing a projection of the light ray in plan along the edge of the ruler, we find point F 3 on the axis OH as a projection of the vanishing point of horizontal projections of light rays (S l F 3 \\ KN).

Let's consider the construction of falling shadows on the steps of the stairs from the side wall (Fig. 113). When constructing shadows in perspective from a building, they usually take the direction of the rays parallel to the picture plane, in this case the rays and shadows from the vertical lines will be parallel, the latter facilitates the construction of shadows in the drawing.

To construct a falling shadow from the side wall of the staircase on the steps, we used the technique of extending the edge from which the shadow is constructed (in this case, the edge A B), until it intersects with the edge on which the falling shadow is constructed.

First we build a shadow from a vertical line A 0 A 1 . for this from the base A 0 We project the beam S 0 to the riser of the first step, at the base of which the shadow breaks and. as from the vertical, on a vertical plane it will go up to the tread. Having reached the second riser, the beam breaks again and rises vertically to the second step, then along the tread the beam will go in the direction of the projection of the beam S 0 until it meets the beam S at the point TO.

Now we build a shadow from an inclined one A B, for this we continue straight A 1 IN" to the intersection with the line IN 1 WITH 1 . belonging to the upper platform R. Shadow from the line A IN 1 at the point 1 will be equal to zero, and the straight line 1-B R will provide shade on the site R from IN to the point 4. To find the shadow on the tread N, let's continue A 1 IN 1 to the point 2, lying in the plane N. and look for the shadow of the point in the same plane IN 1 - this will be a point IN N . When connecting the dots 2 And B N the straight line will intersect the riser N at points 5 And 6. Point 7 on the tread M it turns out the same way. The shadow on risers II and III will be obtained by connecting the points 7 With 6 and 5 s 4.

Shadow from the line IN 1 WITH 1 , so from a horizontal straight line to a horizontal plane it will lie in the direction of the ray going to the same vanishing point as from the point IN R to the vertical wall, from where the shadow will go to point C 1. The remaining constructions are clear from the drawing.

Figure 114 gives an example of constructing falling shadows with rays parallel to the picture plane.