3D Theory - Garland-Heckbert Example


zero error:

error = vT [Q] v

error =

p.x p.y p.z 1
a*a a*b a*c a*d
b*a b*b b*c b*d
c*a c*b c*c c*d
d*a d*b d*c d*d
p.x
p.y
p.z
1
p.x p.y p.z 1
p.x *a*a + p.y * a*b + p.z * a*c + a*d
p.x *b*a + p.y * b*b + p.z * b*c + b*d
p.x *c*a + p.y * c*b + p.z * c*c + c*d
p.x *d*a + p.y * d*b + p.z * d*c + d*d

error = p.x * a * (p.x *a + p.y * b + p.z * c + d) + p.y * b * (p.x *a + p.y * b + p.z * c + d) + p.z * c * (p.x *a + p.y * b + p.z * c + d) + d * (p.x *a + p.y * b + p.z * c + d)

error = (p.x *a + p.y * b + p.z * c + d) * (p.x *a + p.y * b + p.z * c + d)

substituting:

a = n.x
b = n.y
c = n.z
d = -(p dot n)

gives:

error = (p.x *n.x + p.y * n.y + p.z * n.z -(p dot n)) * (p.x *n.x + p.y * n.y + p.z * n.z -(p dot n))

error = (p.x *n.x + p.y * n.y + p.z * n.z - p.x *n.x - p.y * n.y - p.z * n.z) * (p.x *n.x + p.y * n.y + p.z * n.z - p.x *n.x - p.y * n.y - p.z * n.z)

cancelling out terms gives:

error = 0 * 0

error =0

so as required if no edges are collapsed the error is zero


metadata block
see also:

 

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

cover 3D Math Primer - Aimed at complete beginners to vector and matrix algebra.

 

Commercial Software Shop

Where I can, I have put links to Amazon for commercial software, not directly related to this site, but related to the subject being discussed, click on the appropriate country flag to get more details of the software or to buy it from them.

cover EOVIA Carrara Studio 2 (Windows) - This is a commercial 3D modelling tool with some Physics simulation. I think it is aimed at games and animations, not for accurate physics simulation. Eovia (http://www.eovia.com).

See carrara site for information about version 3 launch in Sept 2003.

cover Carrara 3D Basics - A simpler low cost version

 

This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2017 Martin John Baker - All rights reserved - privacy policy.