Affine Interpolation

[Note: The JSP figure has an error so that c is not the value it should be. The original GSP Sketch computes the value correctly. So to get the correct value, download the .gsp file here.]

Affine ratios determine intermediate values at C of affine function z(C) = c, if z(A) = a and z(B) = b are known.

The affine ratio t for C on line AB can be computed from either the x or y coordinate as AC/AB. In this case, there is a vector equation for C:

C = (1-t)A + tB

Then if z is an affine coodinate of a parallel projection of a plane containing AB, or if z is some other affine function, then the value of z at C can be computed from a = z(A), b = z(B) and the ratio t by

z(C) = (1-t)z(A) + tz(B)

c = (1-t)a + tb

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Experiments to try with this figure (with Line AB showing, Values hidden)

First, move C to any position you like. Observe all the ratios that equal t. C does not have to be on the segment AB, you can drag it outside the segment or inside the segment.
Then, drag A and/or B around. Notice that C keeps the same ratio and thus all the other equal ratios also stay the same.

Experiments to try with this figure (with Line AB hidden, Values showing)

Keep C in the same position as before, but drag one or both of the endpoints of the vertical red segments on the left to change the values of a and b. Try both positive and negative values of a and b.
Observe all the ratios that equal t stay the same and that c also stays the same, no matter what a and b are. See the relationship between, a, b, c and the points A3, B3, C3.
Conclude that if the ratio t for C is fixed, even if the points A and B vary and even if the values a anb ad vary, the ratio (c-a)/(b-a) = AC/AB = t stays fixed also.
Repeat the experiment for special values of t, such as t = 1/2, 1/3, 2, -1, -2 to get a feel for what the variou ratios look like.

James King, 1/13/2003