# The Mathematics of Perspective Drawing

Painters use different techniques to create realistic paintings. Vincent Van Gogh’s *Flower Beds in Holland* is one of the examples of such. Given the right distance, you would mistake it for a faded photograph.

But how do painters put so much life in their works? How do they put real space (3 dimensions) on a flat surface (2 dimension)?

One of the techniques used by painters is called **perspective**. Perspective is the art of drawing objects in such a way so as to give them depth and show their distance from the observer. This gives a drawing the illusion of three-dimensional space.

In one-point perspective, the drawings guided by the geometry of converging lines. The lines meet at a point called the *vanishing point* as shown in the second figure. Van Gogh used this technique in the painting above.

The mathematics of perspective drawing lies on the ratios of objects drawn. In one-point perspective, objects are proportional (read Introduction to Similarity for a detailed mathematical explanation). It means, as shown below, that the quotient , where is the height of the tree and is the distance from the vanishing point is always the same. That is, if we make the following representations

: height of the “tallest tree” in the drawing

: distance of the “tallest t tree” from the vanishing point

: height of the middle tree in the drawing

: distance of the middle tree from the vanishing point

: height of the “shortest tree” in the drawing

: distance of the “smallest tree” from the vanishing point

then

.

Note that I placed the quotes on largest tree and shortest tree because if the trees were real, they would have the same height.

One-point perspective is just one of the many techniques in perspective drawing. There are also *two-point perspective*, *three-point perspective*, and *atmospheric perspective*. The combination of these techniques creates realistic and life-like drawings.