Three dimensional space

1
Three dimensional space

• We need to model the space we will be projecting
  onto our canvas.
• We will locate points using x, y, and z
  coordinates using a nonstandard system.
• Introduce the coordinate planes.
• Think of x, y, and z as out, up, and over
  and plot the points (1,2,3), (-2,-1,-3), and
  (0,0,-4).

2
Example

• A rectangular box with faces parallel to the
  coordinate planes has corners at (1,2,1),
  (4,2,1), (1,6,1), and (1,2,8). Find the other
  corners.
• Practice same problem except the only
  information you have is that two corners are at
  (1,1,1) and (6,7,3).

3
Projection
Our goal given the point (x,y,z), how do we find
(x,y)?
4
Similar Triangles

• Two triangles are similar if they have congruent
  corresponding angles.
• Example
• Triangles ACE and BCD are similar.

5
Similar Triangles

• The ratios of corresponding sides of similar
  triangles are equal.
• Example

6
Practice

• Find the missing lengths in the below pictures

7
Finding x and y
Derive the formulas for x and y.
8
Example

• Suppose the viewing point is at (0,0,-3) (so the
  viewing distance is d 3). Find the values of
  x and y for the point (1,1,1).
• What happens as the z coordinate increases?