Descriptive Geometry/Shades & Shadows

SHADOWS in an AXONOMETRIC PROJECTION

To find shades and shadows in an axonometric view, there are two major methods. In the first method, first find the shadows in the top and front views as usual, after constructing the axonometric view, and then transfer them via transfer distances into the final view. 1. Construct top & front view. 2. Construct an axonometric view; in this case, an isometric. 3. Construct the shadows in top view. 4. Transfer the shadows to the axonometric view. If this method becomes too complex because of the many views, simply transfer the light ray into the final axonometric view and use piercing points from that view instead.

Finding Sun’s Azimuth/Angle

Given azimuth α and altitude β for a family of parallel light rays, in order to find the front view of a ray in the family, first you must draw the ray in the top view at angle α from North. Then, using two points on the ray, X and Y, assuming Y pierces a horizontal plane in edge view when seen in the front view, draw a horizontal line through X. Assuming X lies on the folding line, draw a circular arc with radius XY and center X, and name the point where the arc intersects the horizontal line point Z. Project Y and Z onto the edge view in the front view and draw a line from Z that forms angle β with the edge view. Where this ray intersects the projection line from X is X in front view. The ray through X and Y in front view is the front view of the sun’s ray.

Shadows on Curved Surfaces

Given top and front views of an object and a curved wall behind it, extend points of object at the slope of the given cut line and continue it until it hits the surface. Take those points and extend them into the front view. Then extend the points of the object in front view at the slope of the given cut line for front view and continue it until it intersects the second line. That is the point as a shadow. PRACTICE PROBLEMS: Problem 1

Answer to Problem 1

Problem 2

Answer to Problem 2

Objects Casting Shadows on themselves or other objects

When determining shadows of objects, an important factor to consider is overlap of objects and the effects of overlapping objects on a shadows position. If the object casting the shadow or any other objects fall in the path of the shadow, the construction becomes slightly more difficult. The first step is to construct the shadow had it not been interrupted. This will give you the parts of the shadow on the ground plane. Assuming the shapes interrupting the shadow are orthogonal, project the edge lines of the shadow straight up along any vertical surfaces that the lines directly intersect. On the top of these objects, you must now map out the piercing points of the shadow on the faces of the other objects (refer to Piercing Points for more information). Once you have enough points on the ground plane and interrupting objects, connect the points and you have your shadow.

Shadow Projections

Shadow projections are basically piercing point constructions that create a view of an object in space from a certain view. Draw a trace line parallel to the ray line from the point casting the shadow, and mark two points where it intersects the sloped plane in that view. Project those two intersection points downward to the other view. Connect the two points in the other view with a view, this is the trace line. Draw a line parallel to the ray line through the point casting the shadow. Where the ray line intersects the trace line is the piecing point of the point casting the shadow onto the sloped surface. Use this process for every point that falls onto the sloped surface and connect them to create your shadow’s edge. When a shadow no longer falls on the sloped surface then the edge of the shadow will change direction at the point where the surfaces intersect. There are two ways to solve this problem. First create an arbitrary point on the object’s edge in question. Use a piecing point construction to find this point’s shadow and whichever surface it falls on, use the other point that is on that surface and connect those two points and extend the line until it interests the crossing between the two surfaces. The intersection point between the extended line and the two surfaces is the point the edges of the shadows intersect. Another way to do this is find one of the edges of the shadows in point view and project that point of the edge of the surface to another view. Since you had the edge in point view will show you the direction of the edge line. With the direction of the edge of the shadow known you can find the intersection point of the two edges on the intersection line of the two surfaces.

Objects Casting Shadows on Themselves or Other Objects

When determining shadows of objects, an important factor to consider is overlap of objects and the effects of overlapping objects on a shadows position. If the object casting the shadow or any other objects fall in the path of the shadow, the construction becomes slightly more difficult. To do so:

   1. Construct the shadow had it not been interrupted. This will give you the parts of the shadow on the ground plane. 
   2. Assuming the shapes interrupting the shadow are orthogonal, project the edge lines of the shadow straight up along any vertical surfaces that the lines directly intersect. 
   3. On the top of these objects, you must now map out the piercing points of the shadow on the faces of the other objects (refer to Piercing Points for more     information). 
   4. Once you have enough points on the ground plane and interrupting objects, connect the points and you have your shadow.

Projecting Shadows Onto Sloped Planes

Shadow projections are basically piercing point constructions that create a view of an object in space from a certain view.

   1. Draw a trace line parallel to the ray line from the point casting the shadow, and mark two points where it intersects the sloped plane in that view. 
   2. Project those two intersection points downward to the other view. 
   3. Connect the two points in the other view with a view, this is the trace line. 
   4. Draw a line parallel to the ray line through the point casting the shadow. Where the ray line intersects the trace line is the piecing point of the point casting the shadow onto the sloped surface. 
   5. Use this process for every point that falls onto the sloped surface and connect them to create your shadow’s edge.

   P.S. When a shadow no longer falls on the sloped surface then the edge of the shadow will change direction at the point where the surfaces intersect. There are two ways to solve this problem. 
   1a. First create an arbitrary point on the object’s edge in question. 
   2a. Use a piecing point construction to find this point’s shadow and whichever surface it falls on, use the other point that is on that surface and connect those two points and extend the line until it interests the crossing between the two surfaces. 
   3a. The intersection point between the extended line and the two surfaces is the point the edges of the shadows intersect. 
   1b. Another way to do this is find one of the edges of the shadows in point view and project that point of the edge of the surface to another view. Since you had the edge in point view will show you the direction of the edge line. With the direction of the edge of the shadow known you can find the intersection point of the two edges on the intersection line of the two surfaces.

"Projecting Shadows of Curved Surfaces"

Unlike straight surfaces, the shadows of these curved surfaces require more projection points to construct.

Curved surfaces must be constructed using all points of the surfaces since they have no direct endpoints.

Depending on what the curved object is, the method will change.

The first object we would construct a shadow of is the cylinder.

1.Locate the axis of the cylinder in both top and front view.

2.Use the sun angle given and project it down to the floor plane in the front view.

3.Project that point on the floor plan up to the top view and intersect the sun angle projection to form the location of the axis.

4.Construct a circle with the same radius of that circle.

5.You must do this for the bottom circle of the cylinder if it is not touching the floor plane and connecting them.

6.Locate the point in which the sun angle is tangent to the cylinder in top view.

7.Project it down to the front view to form the shadows on the cylinder.

The second curved object is a cone. The cone is pretty simple to construct.

1. Locate the tip of the cone and project it down to the floor on both top and front views.

2. Locate the tangent of the sun's ray to the base of the cone.

If the base of the cone is sitting on the floor, you may connect the tangent to the

projected tip of the shadow.

3. If not, we would use the same rule in projecting a cylinder by location center of the base

and project it to form a circle on the floor plane.

4. We would then connect the tangent of the circle shadow to the tip of the shadow.

5. To construct the shadow on the cone, we would merely connect the tangent to the tip of the cone to form the outline of the shadow.

If the curved surface does no consist of a circle, you can usually use the first method of constructing shadows, by using multiple points on the surface.

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