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| by: O. Bottema, B. Roth |
Theoretical Kinematics
by: O. Bottema, B. Roth
Publisher: Dover Publications
* Number Of Pages: 558
* Publication Date: 1990-09 Dover Reprint, ORIGINAL: 1979
* ISBN-10 / ASIN: 0486663469
* ISBN-13 / EAN: 9780486663463
* Binding: Paperback
Contents
EUCLIDEAN DISPLACEMENTS 1
Orthogonal matrices 3
unit vectors, matrix, mutually orthogonal
The eigenvalues of an orthogonal matrix 4
orthogonal matrix, eigenvalue, invariant plane
Standard representations of a displacement 7
eigenvalue, orthogonal, fixed points
Cayleys formula for an orthogonal matrix 9
orthogonal matrix, Cayley's formula, skew matrix
A displacement as the product of two reflections 11
subspace, orthogonal, displacement
Indirect displacements 12
matrix with determinant, l)-space, m-space
Transformations and operators 13
displacement operator, coordinate system, coordinate transformation
INSTANTANEOUS KINEMATICS 17
The Taylor expansion of an orthogonal matrix 18
skew matrix, inverse motion, symmetric
The angular velocity matrix 20
2m space, moving space, velocity distribution
The case n 23
angular velocity, three-dimensional space, screw motion
Canonical systems 27
screw axis, screw motion, Pliicker vectors
Geometrical invariants 31
screw axis, angular velocity, instantaneous invariants
TWO POSITIONS THEORY 35
Homologous points 36
screw displacement, Y sin y, reflection with respect
The complex of lines joining homologous points 37
homologous points, join A,A2, YM tan
Normal planes 39
tetrahedral, isotropic, null-plane
Homologous planes 41
homologous planes, homologous points, point in E
The midpoint plane 43
displacement vector, linearly dependent, midpoint plane
Homologous lines 45
homologous lines, point H, y sin f
The linear complex P 47
plane a(R, dXW2, homologous lines
The midpoint line 48
oriented line, homologous, midpoint line
The linebisectors 51
linear congruences, coplanar, external bisector
The instantaneous case 55
bisector, torsal line, intersecting Ox
Another method to derive the screw displacement equations 56
displacement vector, origin of coordinates, Rodrigues
THREEPOSITIONS THEORY 63
The plane at infinity 68
geometry, perpendicular, rotations about parallel
The planes through three homologous points 70
kinematics, linear functions, linear transformation
Collinear homologous points 72
twisted cubic, cubic surface, collinear
Lines through three homologous points 74
twisted cubic, ruled surface, point at infinity
The intersection of three homologous planes 76
Ruled surfaces, parabola, triads
Three homologous lines 78
isogonally conjugated, ruled surface, homogeneous function
The circle axes 91
circle axis associated, tangents, tetrahedral complex
The radius of a circle 94
radius, cubic, circle
An oblique coordinate system 96
parallelepiped, coordinate system, cosy
Applications 99
plane at infinity, Plucker coordinates, caplsinp
Three mutually orthogonal screw axes 102
tetrahedron, parallelepiped, associated circle axis
Three screw axes with a common perpendicular 106
infinitesimal, tetrahedral, pure rotation
The instantaneous case 111
osculating plane, three consecutive positions, inflection tangent
FOUR AND MORE POSITIONS 113
The configuration of the screw axes 114
linear congruence, quadrilateral, axes form
The screw cone 117
congruence, vector parallel, angle
The correspondence between axis direction and rotation angle 118
screw cone, cubic cone, rotation angle
The distance condition 119
Plucker vectors, line vectors along, x s/k
The screw congruence 121
fifth axis, quadrilateral, direction s,k
Homologous points 125
plane at infinity, cubic surface, cubic functions
Homologous planes and lines 128
hyperboloid, sextic, homologous quadruples
A special case of four positions 132
circumcenter, dihedral angle, tetrahedron
More than four positions 135
triangle geometry, sextic
The instantaneous case 140
homogeneous coordinates, osculating, points at infinity
CONTINUOUS KINEMATICS 147
Studys soma 150
quaternions, soma, numbers
Eulerian angles 153
sin 6 cos, line of nodes, Eulerian angles
Oneparametermotions 156
Chapter XIII, Chapter X, parameters
The axodes 157
axodes, tangent plane, quadric
The pointpaths 162
osculating plane, binormal, quadratic functions
The motion of a plane 173
torse, path tangent, complex 7.5
The motion of a line 177
tangent plane, torse, striction point
A generalization of the EulerSavary formula 186
Pliicker coordinates, h sin, Euler-Savary formula
Some general remarks 189
centrodes, axode, center of curvature
The velocity distribution 191
paraboloid, velocity vectors, velocity is directed
The acceleration distribution 193
acceleration vector, ellipsoid, acceleration axes
SPHERICAL KINEMATICS 199
The lines joining homologous points 200
isomorphic, non-Euclidean, join P,P2
Three positions 201
trihedron, spherical kinematics, cubic cone
Four and five positions 203
spherical kinematics, rotation axes, four positions
Instantaneous spherical properties 206
geodesic curvature, osculating circle, spherical coordinate
Third order properties 211
polhode, four consecutive positions, Third order properties
Fourth order properties 214
third order, fourth order, z-axis
Time dependent motion 217
normal component, acceleration axes, along the sphere
PLANE KINEMATICS 219
Two positions theory 221
follows from E, pole, triangles
Three positions theory 223
isogonally conjugate, circumcircle, orthocenter
Four positions theory 235
orthocenter, concyclic, optical distance
Five positions theory 249
parabola, concyclic, line at infinity
Six and more positions 255
conies, lines in E, 15 Ball points
Continuous displacements 257
arc length, differential equations, parametric representation
Instantaneous geometric kinematics 263
instantaneous invariants, instantaneous properties, canonical coordinate systems
Second order properties 267
centrode, inflection point, circumcircle
Third order properties 279
line at infinity, circling-point curve, radius of curvature
Fourth order properties 284
arc length, quadratic equations, Burmester points
Translations 290
hyperbola, T-position, pole tangent
Time as a parameter 295
polar coordinates, acceleration pole, second acceleration
SPECIAL MOTIONS 299
The FrenetSerret motion 301
moving frame, trihedron, space curve F
Oarbouxs motion 304
ellipse, plane curves, moving axode
Mannheims motion 310
quartic, Darboux motion, Mannheim motion
Schoenflies motion 312
Mannheim motion, Darboux motion, Schoenflies
Analytically determined motions 315
special motion, inflection line, cubic motion
Linesymmetric motions 317
isotropic lines, cylindroid, skew lines
Planesymmetric motions 329
plane-symmetric motion, planes U(f, series of positions
Motions defined by geometrical conditions 330
cubic parabola, fixed axode, basic developable
Special spherical motions 331
spherical motion, spherical coupler motion, moving axode
Special plane motions 334
cardioid, double points, ellipse
nPARAMETER MOTIONS 355
First order properties 360
MANNHEIM, characteristic point, congruence of normals
Second order properties 364
sinasinA, Gaussian curvature, principal curvatures
Continuous twoparameter spatial motions 373
surface of revolution, quartic surface, Gaussian curvature
Twoparameter spherical motions 379
cyclide, sin2, surface F
Twoparameter plane motions 380
hodograph, pole velocity, zero-position
Examples of threeparameter motions 385
Steiner surface, non-singular
A MAPPING OF PLANE KINEMATICS 393
The geometry of the image space 397
elliptic geometry, linear transformation, null-lines
Straight lines in the image space 401
image space, kinematical meaning, rotation angle
Curves in the image space 403
algebraic curve, point of E, image curve
Plane image curves 407
plane curve, curve k, plane image curve
Quadratic motions 409
The general fourbar motion 413
cross-ratio, elliptic functions, quartic curve
Folding fourbars 418
hyperbola, four-bar, coupler curves
The slidercrank motion 423
cross-ratio, space curve, image curve
Elliptic motion 425
slider-crank, genus, k degenerates
The inverse of a fourbar motion 427
special hyperboloid H, V2 respectively, constrained to move
Cycloidal motions 428
algebraic curve, cycloidal motion
The inverse motion 432
twisted cubic X, k with respect, direct motion
The centrodes 434
fixed centrode, curve k, tangents of k
Instantaneous kinematics 436
instantaneous properties, curvature, image space
Motions with two degrees of freedom 437
image surface, surface F, order N
Algebraic motions with two given paths 441
image surface, algebraic path, accessible region
The mapping of indirect displacements 442
indirect displacement, image plane, displacement D
KINEMATICS IN OTHER GEOMETRIES 447
affine geometry, elliptic geometry, invariant planes
SPECIAL MATHEMATICAL METHODS IN KINEMATICS 501
isotropic coordinates, dual quaternion, Cayley-Klein parameters
BIBLIOGRAPHY 533
INDEX 545
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