To project a three-dimensional image onto a two-dimensional canvas, we will first impose a coordinate system. We will assume the artist's viewpoint O is on the negative z-axis, while the canvas fills the xy plane (also called the picture plane PP). The subject of our painting lies beyond this plane. (Note: yes, we are using a left-handed axes system here.)
We project a point P onto the canvas by creating the line segment OP. Where this segment intersects the picture plane, we have the image of the point, denoted by P'.
Suppose that O has coordinates (0,0,-d) while P has coordinates (x,y,z). We know that P' will have coordinates (x',y',0) and want to find formulas for x' and y'. The figures on the left show top and side views of the original image and its projection. By using similar triangles, we can find the following formulas:
Since we will be concerned with using an image to find an artist's viewpoint (as opposed to using a viewpoint to create an image), we will not be using these formulas later in the site. We will, however, create analogies of them in Flatland.