Re: area of a triangle determinant
- From: lrudolph@xxxxxxxxx (Lee Rudolph)
- Date: 4 Oct 2005 17:18:20 -0400
"vsgdp" <spam@xxxxxxxx> writes:
>My computational geometry book shows how you can find the area of a triangle
>defined by 3 vertices using the cross product.
>
>It then shows that you can also write it by the following determinant:
>
>row 1: (a0, a1, 1)
>row 2: (b0, b1, 1)
>row 3: (c0, c1, 1)
divided by 2!
>I understand how he shows the determinant and cross porduct way are the same
>by just computing it out. But it doesn't really give any insight to how the
>determinant works. He basically says, I have this determinant and it equals
>the cross product, so they are the same. But how would you construct the
>determinant way in the first place? Or did someone just play around and
>made this observation?
The following explanation assumes you know the effects of row
operations and column operations on determinants, as well as the
interpretation of a 3x3 determinant as the volume of a parallelopiped.
The triangle with vertices (a0,a1), (b0,b1), and (c0,c1),
in the Euclidean plane with Cartesian rectangular coordinates,
is congruent (by a translation) to the triangle with vertices
(0,0), (b0-a0,b1-a1), and (c0-a0,c1-a1), so it has the same
(oriented) area; meanwhile, the matrix you gave (by rows)
has the same determinant as the matrix with rows
(0,0,1)
(b0-a0,b1-a1)
(c0-a0,c1-a1)
because the second matrix can be obtained from the first
by performing row operations (subtract row I from row II and
from row III) followed by column operations (subtract a0 times
column III from column I and a1 times column III from column II).
Therefore the formula is correct for the original triangle
if and only if the determinant of the new matrix is the
twice the area of the translated triangle. But this determinant
is, the (oriented) volume of the parallelopiped with edges
(0,0,1), (b0-a0,b1-a1), and (c0-a0,c1-a1), and this parallelopiped
can be dissected into two copies of a rectangular prism with
base the translated triangle and altitude 1. QED
Lee Rudolph
.
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