Math 444 Lab 2, October 10, 2001

Exercise 1.  (an extension of the homework).  Draw 4 points A, B, C, D and construct ALL segments connecting pairs of these points (how many segments?).  Then construct all the midpoints. 

• How many parallelograms can you form with vertices chosen as four of these midpoints?  Is each a midpoint parallelogram of some quadrilateral?
• Choose one of A, B, C or D as a vertex, then add 3 midpoints in a way to form a parallelogram.  How many parallelograms of this kind are in the figure?

Exercise 2.  Areas as barycentric coordinates. In a new sketch, draw 4 points A, B, C, D.  This time plane D inside the triangle ABC.  Construct the LINES (not the segments AD, BD and CD and let the intersection of each line with the opposite side of the triangle be A’, B’, C’.

• Give a convincing explanation why the ratio of area(AA’B)/area(AA’C) = A’B/A’C.
• Then give a convincing explanation why the ratio of area(ADB)/area(ADC) = A’B/A’C.  Check this my measuring the areas and computing the ratios in Sketchpad.
• Use this last fact to prove that if we place masses equal to area (ADB) at C, area(BCD) at A, and area(CAD) at B, the center of mass is D.  In other words, if we let a = area(BCD)/area(ABC), b = area(CAD)/area(ABC) and c = area(ADB)/area(ABC), then a, b, c are barycentric coordinates of D.

Exercise 3

Quadratic Bezier Curves by the De Casteljau Algorithm

We will construct what is called the quadratic Bezier curve by means of a geometric algorithm.

• First, draw segments AB and BC and relabel as A1, A2, A3. (Discover the RELABEL command in Display if you have not done so; it will save you time.) Construct the point B1 as a point on segment A1A2.
• Measure the ratio A1B1/A1A2. We will think of this ratio as t, so you may want to repeat the measurement and relabel it as t as in the figure.

• Select the ratio t and choose Mark Scale Factor in the Transform menu. Then Mark A2 as center and dilate point A3 to get A3', which we relabel as B2.
• Next construct segment B1B2 and construct point C1 on the segment by dilating B2 with center B1 and ratio t.
• Now if you drag B1 all the points B2 and C1 should move "in tandem" with the same barycentric coordinates on their respective segments. You can create an animation button for B1 on segment A1A2 if you like.
• Now construct the locus of C1 by selecting C1 and B1 and choosing Locus on the Construct menu.

• Finally, hide the points B1,B2,C1 and segment B1B2 and the ratio. Select the figure, including the locus and Make a Script. Check in a new Sketch that this figure constructs the locus. In a new figure draw points A, B, C, D, E and construct two Bezier curves given by A, B, C and also C, D, E. Notice that they do not fit together very smoothly unless B, C and D are collinear.
• Write down the formula in terms of the barycentric coordinates (1-t) and t of points B1, B2 with respect to the triangle A1A2A3. What are the barycentric coordinates of C1? Do they add up to 1?

Cubic Bezier Curves by the De Casteljau Algorithm

We will construct what is called the cubic Bezier curve by taking the previous construction a step further.

• Draw points A1, A2, A3, A4 in a new sketch and construct point B1 as a point on segment A1A2.
• Measure the ratio A1B1/A1A2 as before and call it t. Mark Scale Factor as t.
• With this notation, B1 = (1-t)A1+tA2. Construct the points B2 = (1-t)A2 + t A3 by dilating A3 with center A2 and ratio t. Construct B3 on segment A3A4 in the same way.

• Now you have points B1, B2, B3 that all move according to ratio t. But B1, B2, B3 forms a figure like the A1, A2, A3 of the quadratic case. So complete the construction of new points as you did in that case, to get C1, C2 and also D1, all with dilations using the same ratio t but different centers.

• Finally, select point D1 and point B1 and choose Locus from the Construct menu. If you hide the construction and the ratio, you can select the locus and the points A1, A2, A3, A4 and make a script that will draw the locus. Reorder the Givens if necessary to have the order A1, A2, A3, A4. (Do this by dragging in the Script window while you hold down the Shift Key.

• Now apply the script to see how cubics are used in computer graphics. In a new sketch, draw points P1, P2, Q1, Q2, R1, R2. Construct Q2' by rotating Q2 by 180 degrees with center Q1. Now apply the script first to P1, P2, Q2', Q1 and then Q1, Q2, R1, R2. See how the cubics fit together.