Download Perspective Drawing

Perspective Drawing
Discussion
If you look along a straight road, the parallel sides of the road appear meet at a point in the
distance. This point is called the vanishing point and has been used to add realism to art
since the 1400's in Florence, Italy (via city.net).
Suppose you want to draw a railroad track that vanishes into the distance. The rays from the
points a given distance from the eye along the lines of the tracks are projected to the eye.
The angle formed by these rays decreases with increasing distance from the eye. The
picture below shows an overhead view of an observer (camera or eye) looking down the the
track.
The next picture shows a side view. The observer's eye or camera is above the ground.
Draw these pictures on graph paper and try to figure out where the points would fall on the
plane of the drawing. Can you draw the railroad track?
Solution
To draw in perspective, draw a horizon line and draw a vanishing point anywhere on the
horizon. Lines which are parallel in real life are drawn to intersect at the vanishing point.
Distant figures appear smaller but have the same shape and proportions as they would close
up. In geometry, we would say that the figures are similar.
The picture below shows a long hallway with a window in the left wall. The window is a
trapezoid. Can you use your knowledge of geometry to draw another window further down
the hallway? An entire row of windows? To start with the simplest problem, assume the
window tops are all at the same height in the hallway and assume the window bottoms are
all at the same level in the hallway.
Solution to the RR Track Problem
In the view from above we are looking down on the "drawing". The line labelled "plane of
the drawing" might be the top edge of a piece of paper.
In the side view we are looking at the side of the "drawing". The line labelled "plane of the
drawing" might be the right edge of a piece of paper.
For convenience, let's place an x-axis along the bottom of the "drawing" and a y-axis along
the left side. Consider the points, P and Q, where rays from points A and B (on the first
railroad tie) intersect the "drawing" (as shown above). We can measure the values of x1,
x2, and y . . .
. . . and then plot the points (x1,y) and (x2,y). These are the coordinates of the points P and
Q on the "drawing".
Repeat the procedure for the other railroad ties.
Solution to the Hallway Problem
Label the vertices of the first window (trapezoid) A, B, C, and D. We want a second
trapezoid WXYZ which is similar to the first and is drawn in perspective.
The segment AB is the top edge of the first window. The line AB passes through the
vanishing point. The segment WX is the top edge of the second window and the line WX
must also pass through the vanishing point. This is because the window tops are parallel.
Since we choose to make the top edges of both windows the same height, the points A,B
and W,X will all be on the same line which goes through the vanishing point.
The same is true for the bases of the windows.
The difficulty lies in finding the width of the second window. If trapezoid ABCD and
trapezoid WXYZ are similar figures, what can you say about line AC and line WY? Why?
Line AC and line WY are
???
If you have access to a copy of The Geometer's Sketchpad(R), try using it to create a
hallway with windows.