Lab 5: From affine and barycentric coordinates into space
First of all, let's review one of the ways to set up measurements of barycentric coordinates in Sketchpad.
Affine and Barycentric Coordinates
Draw 3 points in a new sketch and label them E0, E1, E2. Draw a fourth point P.
Construct the lines (NOT SEGMENTS) E0E1 and E0E2. Construct the lines through P parallel to E0E1 and E0E2 and construct the intersection points P1 on E0E1 and P2 on E0E2 so that E0P1PP2 is a parallelogram.
Measure the ratios x = E0P1/E0E1 and y = E0P2/E0E2. These are the affine coordinates of P with respect to origin E0 and the axes E0E1 and E0E2.
Experiment 1 "Slope"
For each step of the experiment, do it twice. Do it once with Sketchpad, as described. Then make the corresponding drawing on ordinary graph paper and either really make the construction or do a "thought experiment" based on what you know about ordinary coordinate geometry and algebra.
- In the figure draw a point D and construct line E0D and line E1E2. Let E be the intersection of these two lines.
- Calculate the ratio y/x.
- Animate the point P on line E0D and observe the values of x, y and y/x.
- Compute the ratio EE1/EE2 and compare with y/x. Can you see the relationship and the reason? Hint: Draw lines through E that divide the triangle into 2 triangles similar to E0E1E2 and a parallelogram with sides parallel to E0E1 and E0E2.
- Now construct the line d through D parallel to E1E2 and let the intersections of this line with the axes be D1 and D2. Calculate the ratio DD1/DD2 and compare with the other ratios.
Experiment 2 "Constant Sum"
Calculate the sum x + y.
Observe the sum as you animate P on the line d.
Also observe the sum as you animate P on the line E1E2.
Calculate 1 - x - y and rename this number z. This is defined so that z + y + z = 1. These numbers are the barycentric coordinates of P.
Open-ended explorations
Affine Transformations from a script
Use the script on the server to use two reference triangles to define the affine transformation from P to a point P'. Then use locus to look at, for example, a parallelogram circumscribed around an ellipse.
On the web -- Some space sketches to emulate.
Look at some of the examples at
http://www.cl-gaia.rcts.pt/matematica/sketches/index.htm
Make some sketches like them. Note the use of affine invariant and coordinate ideas.