[Math] Finding an affine combination of a point on a triangle

affine-geometrylinear algebra

I have a problem involving affine combinations that I can't figure out how to solve.

enter image description here

Given the above picture, write q as an affine combination of u and w.

Now, I understand how to write the simpler affine combinations. I can figure out p or s as an an affine combination of u, v, and w. q, however, has me stumped.

I've tried a few different approaches. I started off by looking at the picture using triangles. I wasn't able to get anywhere with this method, given that I couldn't find anywhere where the various laws could be applied.

Next, I tried writing everything as a combination of pretty much everything else, but – after using a number of pages as scrap – I was unable to come up with anything solid. The main problem is that I can't figure out what information will lead me to the answer. I've used variables to represent the distances between q and the other points, but that just gives me equations with too many unknowns.

As for what I know:

$r = \frac{2u + v}3$

$r = \frac{s + p}2$

$r = \frac{3s + w}4$

$p = \frac{2r + w}3$

$p = \frac{s + w}2$

$p = \frac{4u + 2v + 3w}9$

$s = \frac{8u + 4v -3w}9$

$p = \frac{xq + yv}{x + y}$

$q = \frac{p – yv}x$

$q = \frac{aw + bu}{a + b}$

All of these were obtained from the diagram.

  • x is the distance between p and v

  • y is the distance between p and q

  • a is the distance between q and u

  • b is the distance between q and w

A variety of other equalities can be derived from the above relationships, but I've been running myself in circles trying to figure out q in terms of u and w. I would appreciate any help (or a direction to move in) in solving this question!

Thanks in advance!

Best Answer

The key fact here is that $u,v,w$ are affinely independent. Then any point in the plane can be expressed as a unique affine combination of $u,v,w$.

You have $r = \frac{2}{3} u + {1 \over 3} v $, $p = {2 \over 3} r + {1 \over 3 } w$. Combining gives $p = {4 \over 9} u + {2 \over 9} v + {1 \over 3} w$.

We have $q = tp+(1-t) v$ for some $t$, and $t$ is such that the multiplier of $v$ is zero.

So, $q = {4 \over 9}t u + ({2 \over 9}t + (1-t)) v + {1 \over 3}t w$. Setting the multiplier of $v$ to zero gives $t = {9 \over 7}$, and so we have $q = {4 \over 7} u + {3 \over 7} w$.

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