Geometry in Perspective

• Points go to points

• Lines go to lines

• Polygons go to polygons

  

Parallel lines in the world meet in the image "Vanishing Point"

Parallel lines intersect at vanishing point

Line in 3-space

$$x(t) = x_0 + at \\ y(t) = y_0 + at \\ z(t) = z_0 + at \\ $$

Perspective Projection of the line

$$x'(t) = \frac{fx}{z} = \frac{f(x_0 + at)}{z_0 + at} \\ y'(t) = \frac{fy}{z} = \frac{f(y_0 + bt)}{z_0 + at}$$

In the limit as $t \rightarrow \infin$, $x'(t) \rightarrow \frac{fa}{c}$, $y'(t) \rightarrow \frac{fb}{c}$

Vanishing points

• Each set of parallel lines (=direction) meets at a different point

• Sets of parallel lines on the same plane lead to collinear vanishing points. The line is called the horizon for that plane.