We will now investigate some properties of perspective drawing and see how they can help us appreciate art and explore the differences between 2d, 3d, and 4d.

The idea of projective geometry (also known as perspective drawing) is to project a figure onto a plane. For example, we might look at a building outside of a window, and project what we see onto the window with masking tape.

This is an especially effective use of perspective to show depth.

The correct use of vanishing points and other geometric devices can greatly enhance not only one's ability to draw realistically, but also one's ability to appreciate and enjoy art. The picture below shows 2 vanishing points v1 and v2 obtained from the perspective masking tape window drawing above.

Try to identify the vanishing point in the picture below. After you have done this, click on "Here is the answer!" to see the vanishing point (the point in the painting that lines parallel in the real world converge to).

Here is the answer!

Art historian Linda Henderson believes that the discovery of the rules of perspective ushered in the industrial revolution. The newfound ability to reveal depth in drawings led to unprecedented sharing of ideas.

A viewer's eye is located at the point E=(0,0,-d) in the (x,y,z) coordinate system located in 3-space (ie x=0, y=0, z=-d). Notice that just one eye is used. Out in the real world is an object, represented by a vase here. As light rays from points on the object (such as the point P(x,y,z)) travel in straight lines to the viewer's eye, they pierce the picture plane (the x-y plane where z=0), and we imagine them leaving behind appropriately colored dots, such as the point P'(x',y',0). The collection of all projection points P' comprise the perspective image (the perspective drawing) of the object. Take a look at the picture below and re-read this with this picture in mind.

x' = (d x) / (z+d)

y' = (d y) / (z+d)

where d is the distance from the viewer's eye at (0,0,-d) to the picture plane (z=0).

Hence, given a real-life 3D object, the artist will draw x' and y' on their 2D sheet.

I don't expect you to understand how these formulas were obtained, but I do expect you to know that these are formulas for the 2D coordinates on the picture plane of the real-life object that will be drawn in correct perspective (see the picture above).

Suppose the viewer is 3 units from the picture plane. If P(2,4,5) is a point on an object we wish to paint, find the picture plane coordinates (x', y') of the perspective image of P.As a second example, we might want to make a perspective drawing of a real-life Christmas tree. We first put a dot at the image (x',y') of a point (x,y,z) where the coordinates of x' and y' are given by the perspective theorem as above. Then we continue to trace all possible such lines, accumulating all possible points P' associated with our original object. Once we have done this, we will end up with a perspective drawing of our Christmas tree.

SolutionWe have d=3, x=2, y=4, z=5. Thus

x'=(d x) / (z+d) = (3*2)/(5+3)=6/8=3/4 and

y'=(d y) / (z+d) =(3*4)/(5+3)=12/8=3/2.

You will see a chart that is partly filled in with real-life x, y and z coordinates of a house. We will use the viewing distance of 15 to calculate x' and y', and create a perspective drawing of it in Excel. So, we want to mathematically project the three dimensional house onto the mathematically precise perspective image in the plane (where we can draw it). So, we want to transform x, y and z to new coordinates x'=(d x)/(z+d) and y'=(d y) /(z+d).

=d2*a2/(c2+d2)

so type this formula (on the above line) into E2 and hit return. You should now see -1.875. At the bottom right corner of E2 scroll until you get a square with arrows. Then click, hold down, and fill down the Excel column by scrolling down and releasing in E18. The number you will see there is -2.7631579.

Your mouse should now be a thin cross when you take it to your picture.

We want to connect the dots to make the picture represented above.

When a picture is in true one-point perspective, there is an optimal viewing distance d from which we should step back and view it from the vanishing point V.

So, we want to locate the viewing distance d. The way we do this is as follows:

In the drawing of the box below, in order to locate d, we locate V' by drawing the dashed diagonal line of the top face of the box. How do we know that V' is on the same horizontal line as V? Because the dashed lines are images of real lines which are level with the ground, so the sight lines of the viewer to their vanishing points must be level also.

Recall that the reason that we want to determine d is that the picture was drawn assuming that it would be viewed with one eye from a distance d behind V, and so this is the optimal viewing place and distance. To test out our determination of d,

We have only used a little mathematics, but we have accomplished a lot. We have seen the importance of the unique, correct perspective viewpoint. If we view art from the wrong viewpoint, it can appear distorted -- a cube can look like a dumpster. If we view art from the correct viewpoint, it will give us the true perspective desired by the artist. In addition,

Although it is not possible to tell by viewing this small reproduction of

Of course you can't draw lines on the paintings and walls of an art museum, so some other method is needed to find the main vanishing point and the viewing distance. A good solution is to hold up a pair of wooden shish kebab skewers, aligning them with lines in the painting to find the location of their intersection points. First, the main vanishing point V is located. Then one skewer is held horizontally so that it appears to go through V, and the other is held aligned with one of the diagonals of the square tiles; the intersection point of the skewers is then V'. The figures below show people using their skewers to determine the viewpoint of a perspective painting. Then, one by one, the viewers assume the correct viewpoint, looking with one eye to enjoy the full perspective effect. If shish kebab skewers aren't practical, any pair of straight edges, such as credit cards, will work almost as well for discovering viewpoints of perspective works.

Of course there are other important ways to view a painting. It's good to get very close to examine brushwork, glazes, and fine details. It's also good to get far away to see how the artist arranged colors, balanced lights and darks, etc. Our viewpoint-finding techniques add one more way to appreciate, understand, and enjoy many wonderful works of art.