# The Geometry of Metric and Linear Spaces by L. M. Kelly

By L. M. Kelly

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Extra info for The Geometry of Metric and Linear Spaces

Example text

6. TOP VIEW y C(–12,3,99) F picture plane E(0,0,–15) z B(–12,–6,93) A(–12,–18,81) D(–12,–18,117) SIDE VIEW (a) What are the coordinates of the point F ? What are the coordinates of the point G? If the measurements are in inches, how high is the dollhouse? (b) The rest of this problem assumes some familiarity with Microsoft Excel, or some other spreadsheet program. 7(a) and (b) are the xyz-coordinates of all 17 vertices of the Perspective by the Numbers 19 house. In the 4th column is the viewing distance (d = 15 units).

7. 11. How do you finish the cube if the viewing distance is 7 inches? 2. 12 is not a cube. Let’s say its front face is a square, but its top face is in reality twice as long as it is wide from left to right. In this case, the viewing distance is not equal to the distance between the two trees. What is the viewing distance? ) What if the top is three times as long as it is wide from left to right? 12. What if the box is not a cube? 3. Now do the real thing: go to a gallery or museum and practice your viewing techniques!

The perspective image P ′ Q′ of a line segment P Q is also a line segment (unless P Q is seen end-on by the viewer, in which case the image is a point). Rule 2. , lies in a plane parallel to the picture plane) has a perspective image P ′ Q′ that is parallel to P Q. 1. A line segment P Q and its image P ′ Q′ . 1. First, observe that the triangle △EP Q lies in a plane, and the intersection of that plane with the picture plane is the line containing the segment P ′ Q′ . Second, if P Q lies in a plane z = k parallel to the picture plane, then the lines containing P Q and P ′ Q′ cannot intersect, and furthermore, these two lines are coplanar, since they lie in the plane containing △EP Q.