From f7e473b98491a14a44c57f5bc2a9b489784bd7d4 Mon Sep 17 00:00:00 2001 From: Stefan Goessner Date: Tue, 13 Nov 2018 09:39:33 +0100 Subject: [PATCH] Version 2.3.5 --- triangle.html | 41 ----------------------------------------- 1 file changed, 41 deletions(-) delete mode 100644 triangle.html diff --git a/triangle.html b/triangle.html deleted file mode 100644 index 655863e..0000000 --- a/triangle.html +++ /dev/null @@ -1,41 +0,0 @@ - - - - - - - - -

The Right Triangle

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Base Geometry

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Let the right triangle hypothenuse be aligned with the coordinate system x-axis. -The vector loop closure equation running counter-clockwise then reads

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aeα+be~α+cex=0a{\bold e}_\alpha + b\tilde{\bold e}_\alpha + c{\bold e}_x = \bold 0(1)

with

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eα=(cos⁡αsin⁡α)ande~α=(−sin⁡αcos⁡α){\bold e}_\alpha = \begin{pmatrix}\cos\alpha\\ \sin\alpha\end{pmatrix} \quad and \quad {\tilde\bold e}_\alpha = \begin{pmatrix}-\sin\alpha\\ \cos\alpha\end{pmatrix}

Resolving for the hypothenuse part cexc{\bold e}_x in the loop closure equation (1)

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−cex=aeα+be~α-c{\bold e}_x = a{\bold e}_\alpha + b\tilde{\bold e}_\alpha

and squaring

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finally results in the Pythagorean theorem (2)

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c2=a2+b2c^2 = a^2 + b^2(2)
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More Triangle Stuff

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Introducing the hypothenuse segments p=a⋅exp={\bold a}\cdot{\bold e}_x and q=b⋅exq={\bold b}\cdot{\bold e}_x, we can further obtain the following useful formulas.

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segment psegment qheight harea
cp=a2cp = a^2cq=b2cq = b^2pq=h2pq = h^2ab=chab = ch
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