Notes on Perspective Drawing

These are our notes to studying perspective without the crutch of cubes. They had better help our drawing skills. Please let us know any inaccuracies.

The Perspective Projection Map

Let $V$ be the viewing plane and $O$ be the eye point. Then the perspective drawing is,
in essence, the projection to $V$ with respect to $O$. More precisely, the image of a point $P$ is the point of intersection of the line $OP$ and the plane $V$. Let $f$ denote this map.

The Vanishing Point

Consider the images of parallel lines $\ell_\alpha$ for $\alpha$ running over an index set. The intersection of the images of these parallel lines is called the vanishing point. We determine its existence and its location geometrically. Let $W_\alpha$ be the plane that contains $O$ and $\ell_\alpha$. Then the vanishing point lies on the $W_\alpha \cap V$ for each $\alpha$. Thus it lies on $(\cap_\alpha W_\alpha) \cap V$. The first factor $\cap_\alpha W_\alpha$ is obviously the line $\ell$ which is parallel to $\ell_\alpha$ and which contains $O$. Thus the vanishing point is the point of intersection of $\ell$ and $V$ if $\ell$ is not parallel to $V$.

The Transfer of Distance

Let $\ell_0$ be a line parallel to $V$. The perspective projection map when restricted to $\ell_0$ is scaling. Thus it is easy to draw equispaced points on $f(\ell_0)$. We would like a way to transfer distance on $f(\ell_0)$ to that on images of lines not parallel to $V$. Let $P$ and $Q$ be two points of $\ell_0$ and let $\bar{P}$ and $\bar{Q}$ be their images respectively. Let $\ell$ be a line not parallel to $V$ such that it passes through $P$. Let $Q’$ be the point on $\ell$ such that $Q’$ is further away from the eye point and such that the lengths of $PQ$ and $PQ’$ are equal. Let $N$ be the vanishing point for the image of the line $PQ’$, which is just $\ell$, and let $M$ be the vanishing point of the image of the line $QQ’$. Then the image of the line $PQ’$ is the line $\bar{P}N$ and that of the line $QQ’$ is $\bar{Q}M$. Thus $f(Q’)$ is the intersection of the line $\bar{P}N$ and $\bar{Q}M$. It is easy to see that the lengths of $MN$ and $ON$ are equal by observing the isosceles.