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:
- Use a thumb and a forefinger to measure the distance d between V and V' which you should see as 2 dots at the very top of your screen. Keep this distance, called d, with your hand.
- Continuing to keep that distance, turn your hand to measure out d units in front of V (so d is now perpendicular to the screen instead of being inside of it).
- Place your left eye exactly at that same point (d units in front of V - not in the center of the page). You will need to get very close to the computer screen to do this.
- Close your right eye.
- Without changing your position, let your left eye roll down and to the left and then look at the box. Although it may be too close for comfortable viewing because you will be so close to the computer screen, the 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 astigmatism...
- Now switch roles and read the directions to your partner as they complete this activity.
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
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!
.
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.
use many more rows of Excel. The full-body version of
this Yoda uses 53,756 vertices!
