Calculating the Viewing Distance for Interior of Antwerp Cathedral, by Peter Neeffs the Elder, 1651

In the figure below, we see the trick applied to finding the viewpoint for the Interior of Antwerp Cathedral painting. We first determine the vanishing point V directly in front of us, which is easy to see, as it is the intersection of lines which are supposed to be parallel in the real-world. Some of the lines have been drawn in below in order to highlight V. Notice that lines that follow along the edges (coming from us towards V) of the square tiles of the floor also intersect at V. Since the floor tiles are squares, they serve the same purpose as the square top of the cube in the previous discussion. Hence, our second point V' is calculated by following along a diagonal (indicated on the picture) that follows along the vertices of the square tiles. If we had chosen the diagonals of other square tiles, we still would have converged to V', or to a point V'' on the other side of V that is also d units away from V. In either case, the viewing distance d is the indicated length, and the correct viewpoint is directly in front of the main vanishing point V.

Although it is not possible to tell by viewing this small reproduction of Interior of Antwerp Cathedral, the effect of viewing the actual painting in the Indianapolis Museum of Art gives a surprising sensation of depth, of being "in" the cathedral. The viewing distance is only about 24 inches, so most viewers never view the painting from the best spot for the sensation of depth!

To test out our determination of d

  • Close your right eye.
  • Place your left eye directly in front of the point V.
  • Measure the distance between V and V' which you should see as 2 dots at the very top of your screen.
  • Use this to measure out d units in front of V. Move until your left eye is exactly d units away from V on the computer screen.
  • Without changing your position, let your eye roll down and to the left and then look at the box. The rectangular distortion should go away and it should look much more like a cube! Most people will be able to see this by following the directions, but some people may have problems due to glasses, ...

    Using Mathematics to Create Precise Perspective Drawings

    There are precise mathematical rules for perspective drawing and understanding just a little bit about these rules can help us view art more effectively.

    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) on the vase) 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.

    Perspective Theorem

    Given a point (x,y,z) of a real-life object with z > 0, the projections of these real-life 3D vase coordinates onto the 2D sheet (the perspective drawing coordinates) are given by the mathematical formulas.
    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 3-D object, the artist will draw x' and y' on their 2-D 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 2-D 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.


    We 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.

    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.

    Perspective Drawing in Excel

    We are going to make a perspective drawing of a house in Excel using the above equations. Work in a group of 2. 1 person should read these directions to the 2nd person, while the 2nd person actually does this on their computer. You only need to do 1 Excel house per group. Click on this excel file. The computer will download the file onto the computer. Open up Microsoft Excel by clicking on the green X icon on the Dock. From Excel, use File and release on open and then look for the file perspectivehouse.xls on the Desktop or in the Documents folder.

    You will see a chart that is partly filled in with real-life x, y and z coordinates of a house (in columns A, B and C, respectively). We will use the viewing distance of 15 (as in column D) 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). We will make Excel do these formulas for us!.
  • To transform x to x', click on E2 in the Excel sheet (row 2, column E). The Excel formula for x' that you should type in is:
    so type this formula (on the above line) into E2 and hit return. You should now see -1.875. Since d is in the d2 box, x is in the a2 box, and z is in the c2 box, we see that the Excel formula that we just typed in -
    is the correct formula to use in order to find the 2-D perspective coordinates of the given 3-D point, since it corresponds to the formula
    x'=(d x)/(z+d)
    Be sure that you understand this before moving on.
    Click on E2 again. At the bottom right corner of E2 scroll until you get a square with arrows which looks like . 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.
  • To transform y', click on F2 in the Excel sheet. Figure out the Excel formula corresponding to y'=(d y)/(z+d) (hint - first make sure you understand the Excel formula that we used for x', and then figure out the corresponding Excel formula for y') and then type your Excel formula into F2 and hit return. You should now see -2.8125. At the bottom corner of F2 click until you get a square with arrows. Then fill down the Excel column and release in F18. The number you will see there is .39473684.
  • To draw our house, click on the grey E box, so that that column is highlighted. Then hold down the shift key while you click on the grey F box, so that both the E and F columns are now highlighted. Under Insert, release on Chart. Then click on XY (Scatter) and then on Finish. Now we have our mathematical drawing! All we have to do is connect the dots (see below)

  • Click on the blue or grey background part of your chart. Under View, scroll to toolbars, and then release on drawing. A "drawing toolbar" will appear. Click on the line that is just a straight line (no arrow heads or anything).
    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.
  • Click on the bottom left point in the picture, hold down the mouse, and release on the point above it to draw the line representing the left side of the house. Then click on the blue part to complete the line.
  • Keep clicking on the line on the toolbar in order to draw lines on your picture in order to complete the picture as above.
  • To erase a line that you have drawn by mistake, click on the arrow in the drawing tools, and then go back to the picture. Hold the mouse over the line until it turns into a hand. Wait until it tells you Line (--) where (--) means some number. Then click down, and hit delete.
  • To make lines that are not formed directly by connecting 2 dots (the line coming down from the top left of the roof and the little line from the top right of the base of the house connecting to the roof), first you make a complete line down to the corresponding dot, then make a line to the part that you desired, and then erase the longer line.
    Extra Credit Find a real-life painting that was done in one-point perspective and adapt the method to find the viewing distance of this painting. Write up a report on how you did this, giving specific details, and also discuss what you saw when you used the proper viewing distance for your painting. I recommend that you re-read the reading above before you try this.

    Adapted from Marc Frantz's Mathematics and Art.