planeparamofT
by Tim C. Lueth, SG-Lib Toolbox: SolidGeometry 5.6 - Analytical Geometry
Introduced first in SolidGeometry 4.3, Creation date: 2018-09-29, Last change: 2025-09-14
returns the plane parameters from a HT matrix (x/y plane) - ax+by+cz+d=0
Use carefully: The 'planeparamofT' function has not yet been extensively tested
Description
Linear equation system using T=[ex ey ez o]
a b c d =0
o+ex + 1 = 0
o+ey + 1 = 0
o + 1 = 0
==> a b c for d==1
See Also: Tofplaneparam
Example Illustration
Syntax
[pp,ez]=planeparamofT(T,[d])
Input Parameter
Output Parameter
pp: | | plane parameter [a b c d]; ax+by+cz+d=0 |
ez: | | normal vector |
Copyright 2018-2025 Tim C. Lueth. All rights reserved. The code is the property of Tim C. Lueth and may not be redistributed or modified without explicit written permission. This software may be used free of charge for academic research and teaching purposes only. Commercial use, redistribution, modification, or reverse engineering is strictly prohibited. Access to source code is restricted and granted only under specific agreements. For licensing inquiries or commercial use, please contact: Tim C. Lueth
Algorithm (Workflow)
This function, planeparamofT, calculates the parameters of a plane from a homogeneous transformation matrix (HT matrix) T. The plane is defined in the form ax + by + cz + d = 0.
Input Parameters
- T: A 4x4 homogeneous transformation matrix.
- d: An optional parameter, defaulting to 1, which scales the plane parameters.
Output Results
- pp: A vector containing the plane parameters [a, b, c, d].
- ez: The normal vector of the plane, extracted from the third column of T.
Algorithm Steps
- Initialize
d to 1. If a second argument is provided, update d with its value.
- Construct a matrix
VL using the first three rows and specific columns of T. This matrix is used to derive the plane parameters.
- Calculate the rank of
VL. If the rank is less than 3, issue a warning that no solution can be found.
- Use the reduced row echelon form (RREF) to solve the linear system represented by
VL and a column of -1s. This yields the plane parameters.
- Scale the plane parameters by
d and append 1 to form the output pp.
- Extract the third column of T as the normal vector
ez.
- If no output is requested, visualize the plane and transformation using plotting functions.
Algorithm explaination created using ChatGPT on 2025-08-19 01:26. (Please note: No guarantee for the correctness of this explanation)
Last html export of this page out of FM database by TL: 2025-09-21