Math 445-487 Lab 1C Plotting Points

I. Coordinates on a line

Computing affine coordinates on a line

Coordinates on a Line

Draw a line AB and a point P on the line. Then select (in order) A, B, P and choose Ratio from the Measure menu. This ratio should be AP/AB. This is the affine coordinate s of P on the line with origin at A and unit at B.

• Drag P back and forth so that you have a clear sense of where the ratio is positive, where it is negative, where it is less than 1 and where it is great than 1, etc.

One more coordinate

• Now compute the ratio BP/BA the same way. Call this coordinate t.
• Calculate s + t as P moves along line AB.
• What do you find? Why is it true?

We call s and t the barycentric coordinates of P with respect to A and B.

Related coordinates on two lines (parallel projection)

Let line AB above be line m. Draw any line a through A and call it a. Then construct line b through B parallel to a; construct line p through P parallel to a.

Then draw any line n transversal to the parallel lines and define A', B' and P' as the intersections of n with lines a, b, p.

• Check that the ratio A'P'/A'B' equals AP/AB always, in magnitude and sign.
• Check that the barycentric coordinates of P with respect to A and B are the same as the barycentric coordinates of P' with respect to A' and B'.

Plotting Points on a line from Affine Coordinates

In a new sketch draw a line AB.

To plot a point P with affine coordinate 1/3 with respect to AB.

• Mark A as center.
• Dilate B with ratio 1/3.

To plot a point P with a variable affine coordinate s with respect to AB.

• Choose Graph > New Parameter.
• Name the parameter s and type in something like 4/5.
• Mark A as center and Mark s as Scale Factor.
• Dilate B.

Now if you want to change s, just double click on s and type in a new value. See how P move.

Save These 2 Sketches to Turn in Later.

Plotting Points on a Plane from Affine Coordinates

Draw a triangle ABC in a new sketch. Create new paramaters names x and y. Calculate z = 1 - x - y.

Task: Construct a point P with affine coordinates x and y with respect to ABC.

Plotting Points on a Plane from Affine Coordinates

Draw a triangle ABC in a new sketch.

• Construct a figure like this one.
• Explain what are the barycentric coordinates of each point.