In the second column we freed the circle from being a flat-on geometric shape so that it could move out into space as the ellipse. We’ve used it to help us draw a pot and to see the roundness of forms, and now we’re going to use that ellipse to fly us into an imaginary scene that introduces us to the principles of perspective.

We follow that flying Frisbee of an ellipse as it settles down as a perfect little pond on a vast Kansas prairie. A man walks out onto that plain with a picnic basket, a blanket and a beagle. He sits down on his blanket to admire the view and the improbably perfect pond.

The beagle catches the scent of the little rabbit on the other side of the pond and takes off after it. Ignoring the shouts of his master, the dog paddles through the pond, bounds across the vast expanse and disappears over the horizon. (Two nice farmers in the next town find him and call the ASPCA.)

The runaway beagle’s trajectory has given us a vanishing point, the first element in the geometry of perspective: the point on the horizon towards which objects in the picture converge. In the first drawing, the man is sitting down so his viewpoint is low (and let’s imagine that we’re in a slightly elevated position behind him), and because the horizon line occurs roughly at eye level, the horizon line is also low and all the shapes appear relatively flattened out. Also, in one point perspective, all the lines running from left to right are parallel to the horizon line.

In the second diagram, where the man stands up to call his dog, he sees the scene from a higher viewpoint and thus the horizon line is also higher within the rectangle of our image. Now the blanket and the pool become wider, front to back, as does the perceived distance between the man’s feet and the horizon. It’s just as the ellipses in the drawing of the pot became wider in the same way the more we looked down on them.

As useful as one-point perspective is in drawing a Kansas picnic or highways in the Nevada dessert leading straight to the sunset, other scenes require more complicated angles. For these images we need two-point perspective.

Let’s start by going back to the circle and plotting it in two-point perspective so we know how to make an official ellipse. It may not be as fluid or interesting as your free-hand ellipse, but you should know how to do it so you can move on with your life.

Are you still with me? Now, something a little easier to do. Choose two vanishing points, left and right (J and I), along the horizon and roughly equidistant from the center. Now draw lines from the right vanishing point (I) to the top corners of the box and to the intersection points C, A and D. Count the lines you have just made — there should be five.

Now, by drawing a line from vanishing point J to the right corner of the box, we are crossing four lines (check the diagram) that give us important intersections. The first is intersection K , the point at which you can make a horizontal line to complete the perspective square. Think of it as a flap bending away from the bottom box. I’ll get to the second intersection in a minute. The third is intersection L, which shows you where to make another horizontal line to establish the center of the perspective box, and marking both left and right intersections “point B” to match how they are identified in the lower box.

The second and fourth intersections, E and H, along with intersections G, F, B and A, match the same points in the lower box and give you the theoretical means to draw a circle seen in perspective. The theory is that you simply connect these eight points (A and B are doubled) with curved lines and, voila, you have the correct ellipse. However, I find it takes a certain amount of fiddling to swing these curves around the corners to make them look right. In other words, you already have to have some sense of what a perspective circle looks like in order to carry out this last bit of the procedure. Whew! Work on your free-hand ellipses.

Now, back to our Kansas prairie picnic. This time, we’ll let our beagle run off at an angle, which will give us a vanishing point, A, to the right of our picture frame. Establish the left-hand vanishing point, B, along the horizon at roughly the same distance from the center as the first vanishing point (as you did in plotting the perspective circle). Choose a point in the lower left of the picture frame along the angle of the first vanishing point for the corner of the blanket, (C). Now join that point to the second vanishing point. This gives you the angles of two sides of the blanket. Now choose two points that seem reasonable for the width (D), and length (E), of the blanket and join those points to the appropriate vanishing points. Now you have completed the perspective view of the rectangle of the blanket as it has turned to match the trajectory of the beagle’s flight.

Since we’ve spent so much time plotting the circle in perspective (a.k.a. the ellipse), let’s turn our pond into a little house on the prairie to get some practice with rectilinear shapes.

First, choose a point (F) above and to the right of the blanket for the near corner of the house (as you did with the blanket), and extend lines along the vanishing point angles to establish the length (H) and width (G) of the house. From the point F draw a vertical line to establish the height (I) of the house.

Using the vanishing point trajectories you can now complete the basic box of the house. In order to establish the center points of the two visible walls, make a horizontal base line, J to K, running through point F at the corner of the house. Using lines running from both vanishing points and through the corners of the house establish the width and length along the base line, J to K. Measure the halfway points along that line. By extending vanishing point lines back to the house from the two midpoints you can figure out where to put the centered roof peak and the centered door, and where to center windows in the remaining spaces. You have now completed a scene of a blanket and a house viewed from the same vantage point

My personal take on perspective is that one should understand enough of the basic idea of vanishing points to substantiate how you see objects and buildings recede in space in your everyday life, so that it helps you to draw a convincing image without having to do a lot of plotting. An easy little exercise you can do is to draw a rectangle with a horizon line and then, free-hand, draw a series of boxes aligning with the same one or two vanishing points. It will help you, too, with understanding what things look like when they are low or high in an image field.

For those of you anxious to move deeper into the labyrinth of perspective, I offer this little taste of what you’re in for.

But to see what a great artist can create playing around with very simple perspective, I include this painting, “Melancholy and Mystery of a Street,” by Giorgio de Chirico.

*Giorgio de Chirico’s “Melancholy and Mystery of a Street”*

*James McMullan, New York Times*