Class Activities on Perspective Drawing and Projective Geometry

The purpose of the lab component of the course is to extend and build connections to course material in an interactive manner. During labs you will work at your own pace, preferably in groups of 2-3, as I make my way around the room to check in (and help!).

Read and complete the activities as you answer the following questions on a sheet of paper.
  1. First take out your homework notes and get started below. Show your hw to me as I make my way around the room and ask me any questions--I'm happy to help!

    Activity 1: Come One - Come All - to a Better Cube

    A picture drawn in perspective was drawn assuming that it would be viewed with one eye from a distance d behind V, and so this is the mathematically optimal viewing place and distance. To test out our determination of d,

    Click on this link of the large drawing and do as follows as you have a partner read you these directions:

  2. Were you able to see the distortion disappear?
  3. As a review, briefly summarize where was your left eye placed?

    Viewers in a gallery or museum almost never assume the correct perspective viewpoint, because these locations tend to be rather low, off-center, rather close, etc. It's true that the artwork may appear perfectly acceptable from most locations in the room. However, when one assumes the correct viewpoint and looks with one eye, it is common for the illusion of depth to be astonishingly more believable. You can almost "feel" the space! (Of course the work must be of high quality and very precise.) While you can't draw dashed lines on the artwork to find the viewing distance, you can mentally trace appropriate lines by holding up shish kebab skewers. Once you do so, you can use your hands to measure the viewing distance in the painting and stand d units in front of V.

    Activity 2: Using Mathematics to Create Precise Perspective Drawings and Computer Animations

  4. Given a point (x,y,z) of a real-life object with d > 0, write out the mathematical formulas for the perspective drawing coordinates of x' and y' that were given on the homework reading.

    We are going to make the computer create a perspective drawing of a house by using these equations. Work with a partner - have one person read the directions on their screen while the other performs the instructions.

    Microsoft Excel is an electronic spreadsheet program that we'll use to algebraically manipulate and analyze data and create visualizations. Data is organized into boxes or cells that are labeled by their row number and column letter.

    Download and open this Excel file using the program with the green X. 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!
  1. What Excel formula should we use in F2 corresponding to y'=(d y)/(z+d) (for the second row)?

Digital Movies

Digital animations such as use many more rows of Excel. The full-body version of this Yoda uses 53,756 vertices!

Models created by Kecskemeti B. Zoltan and visualized by T. Chartier. Images courtesy of Lucasfilm LTD as on
Using the Force of Math in Star Wars

  1. List the artists and mathematicians who are mentioned in the perspective drawing homework reading and give a very brief summary of their contributions.
    Example: Julian Beever, pavement drawings in perspective
  2. Take the ASULearn Mathematical Experiences reflection
  3. Look over the hw for tomorrow from the main calendar page and ask me any questions before you leave (as time allows).

Adapted from Marc Frantz's Mathematics and Art.