MECH 576
Geometry in Mechanics
Collineation and Cross Ratio in Planar Projective Mapping
November 19, 2009
1
An Exercise
Examine Fig. 1. It illustrates an exercise for freshmen at TU-Dresden wherein t
MECH 576
Geometry in Mechanics
Shortest Distance from Line to Coplanar Conic
November 25, 2009
1
A Conic and Two Tangent Lines
Examine Fig. 1. Shown there is a given conic a, represented by a general
MECH 576
Geometry in Mechanics
November 30, 2009
Kinematics of Clavels Delta Robot
1
The DELTA Robot
DELTA, a three dimensional translational manipulator, appears below in Fig. 1.
Figure 1: Symmetrica
MECH 576
Geometry in Mechanics
December 8, 2009
Instantaneous Twist Axis and Angular Velocity from Velocity of Three Points
1
Method and Axis Direction
Ab initio, one is given, on some rigid body, the
MECH 576
Geometry in Mechanics
September 16, 2009
Grassmannian Reduction of Quadratic Forms
Abstract: Conic sections, i.e., ellipses, hyperbol and parabol, are encountered in elementary
geometry but c
MECH 576
Geometry in Mechanics
September 16, 2009
Inspiration, the Essential Grassmannian and Projectivity
1
Inspiration
Four of my favourite quotes, that pertain to geometric thinking, are written be
MECH 576
Geometry in Mechanics
November 19, 2009
Centre of a Hyperboloid of One Sheet via Line Geometry
1
Introduction
A unique hyperboloid of one sheet may be dened by ruling three given skew lines P
MECH 576
Geometry in Mechanics
September 16, 2009
Line Geometry Primer
0. LINE GEOMETRY
(what is it about?)
Writing constraint equations dealing with lines in 3D space
Writing ecient, test-free algori
MECH 576
Geometry in Mechanics
September 16, 2009
Using Line Geometry
1
Deriving Equations in Line Coordinates
Four exercises in deriving fundamental geometric equations with line coordinates will be
MECH 576
Geometry in Mechanics
September 16, 2009
Pedal Point and Quadric Axes
1
Fupunkt
Consider the following very easy problem. It may be eectively used, by solving it in conventional
and not so co
MECH 576
Geometry in Mechanics
September 16, 2009
Solving Simultaneous Multivariate Polynomial Equations
1
Explicit Solutions of Cubic and Quartic Univariates
Although explicit solutions to third and
MECH 576
Geometry in Mechanics
November 19, 2009
Gram-Schmidt Orthogonalization
and Upper Triangularization
1
Orthogonalization
It is shown how a basis, represented by three column vectors in the 3 3
MECH 576
Geometry in Mechanics
October 28, 2009
Introduction to Conics and Quadrics
1
Conics and Quadrics by Other Means
Second order planar curves and spatial surfaces can be dened on ve and nine poi
MECH 576
Geometry in Mechanics
September 16, 2009
Parametric and Bzier Curves
e
1
Parametric Cubics and Quintics
Before studying the following material, review and absorb the introductory ideas dealt
MECH 576
Geometry in Mechanics
October 21, 2009
The Inertia Matrix in Least-Squares Linear Fitting
ABSTRACT Least squares tting of point sets to lines, planes, curves and surfaces is carried
out using
MECH 576
Geometry in Mechanics
September 16, 2009
Planar Kinematic Mapping Fundamentals
1
Introduction
The intent of this article is to explain, in a clear and simple way, what planar kinematic mappin