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 they must manually
construct, with drafting instruments
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 ellipse, and a given
coplanar line g cfw_G0 : G1 : G2 .
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: Symmetrical (Conventional) DELTA Robot
In what follows, inverse a
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 position vectors of three points A, B, C as a, b, c an
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 chopping cones with planes is regarded as mathematical e
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 below. The second
is from a modern book on classical geom
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 ,Q,R with a
one parameter set of lines that intersects
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 algorithms
Mantra: Computers are fast, memory is cheap
Answer
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 conducted.
These results may be widely used to formulat
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 conventional ways, to reinforce geometric thinking. Such
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 fourth order polynomial equations in one variable exist
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 matrix A, is converted
to the three orthogonal vectors
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 points, respectively.
Now some other, possibly more ecient
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 with in
MECH 289, Design Graphics, Module M2, section 6
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 eigenvalues and eigenvectors to nd the major principal
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 mapping
is in terms of elementary homogeneous matrix transfor