Farin G. Curves and Surfaces for CAGD. A Practical Guide
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442 Appendix A Quick Reference of Curve and Surface Terms
Patch Complicated surfaces are usually broken down into smaller units, called
patches. For example, a bicubic spline surface consists of a collection of bicubic
patches.
Point A location in space. If one uses coordinate systems to describe space, a
point is represented as an w-tuple of real numbers.
Point cloud A (typically large) set of 3D points without any ordering or struc-
ture.
Precision A curve or surface generation scheme has nth order precision if it
reproduces polynomials of degree n.
Projective map A map comprising affine maps and central projections. Leaves
cross ratios of coUinear points unchanged. Does (in general) not map parallels
to parallels.
Quad Short for quadrilateral.
Quadric A surface with the implicit representation f{x, y, z) = 0, where f is
a quadratic polynomial. Or: the projective image of an elliptic paraboloid, a
hyperbolic paraboloid, or a parabolic cyHnder.
Ratio A quantity computed from three coUinear points. Invariant under affine
maps,
but not under projective maps.
Rational curves and surfaces Projections of nonrational (integral) curves or
surfaces from four-space into three-space.
Recursive subdivision Curves or surfaces that are defined as the limit of a
polygon or polyhedron refinement process.
Ruled surface A surface containing a family of straight lines. Obtained as linear
interpolation between two given curves.
S-patch A surface patch with an arbitrary number of boundary curves, con-
structed by mapping a multidimensional simplex onto a 2D polygon, the domain
of the patch.
Segment An individual polynomial (or rational polynomial) curve in an assem-
bly of such curves to form a spline curve. The bivariate analog of a segment is a
patch.
Shape parameter A degree of freedom (usually a real number) in a curve or
surface representation that can be used to fine-tune the shape of that curve or
surface.
Solid modeling The description of closed objects that are bounded by a collec-
tion of surfaces.
Space The collection of all points.
Quick Reference of Curve and Surface Terms 443
Spline curve A continuous curve that comprises several polynomial segments.
Spline curves are often represented in terms of B-splines. They may be the result
of an interpolation process or of an ab initio design process. If the segments are
rational polynomials, we have a rational spline curve.
Standard form The property of a rational curve of having its end weights equal
to unity.
Star In a triangular mesh: the set of all triangles having a given point as a vertex.
Stereo lithography The process of producing a physical (usually plastic) model
of a part, involving building layers of material hardened by laser rays aimed
inside a tank of liquid resin.
Subdivision Breaking a curve or surface down into smaller pieces of the same
type as the original curve or surface.
Subdivision surface A surface that is the result of iteratively refining a given
control mesh.
Support The region over which a nonnegative function is actually positive.
Surface The locus of all points of a moving and deforming curve. Or: the 3D
image of a region in two-space under a continuous map. A surface is often broken
down into patches.
Surface spline A piecewise bicubic approximation to a Doo-Sabin surface.
Surface triangulation A collection of triangular facets that covers a smooth
surface, obeying the structure of a triangulation.
Tangent The straight line that best approximates a smooth curve at a point on
it. This straight line is parallel to the tangent vector.
Tangent vector The first derivative of a differentiable curve at a point on it. The
length of the tangent vector depends on the parametrization of the curve.
Tensor product A method to generate rectangular surfaces using curve methods.
Tile The interior of a convex closed 2D polygon.
Torsion A measure of how much a curve "curves away" from the osculating
plane at a given point.
Transfinite interpolation Interpolating to curves, with infinitely, that is, trans-
finitely many points on it, as opposed to discrete interpolation, which interpolates
only to finitely many points.
Translational surface A surface that is obtained by translating one curve along
another one.
Triangular patch A patch whose domain is a triangle.
444 Appendix A Quick Reference of Curve and Surface Terms
Triangulation A collection of triangles, covering a region in E^, such that the
triangles do not overlap, and that any two triangles either have no points in
common, or they have one edge in common, or they have one vertex in common.
See also surface triangulation.
Trimmed surface If the domain of a parametric surface is divided into "valid"
and "invalid" regions, the image of the valid regions is called a trimmed surface.
Twist vector The mixed second partial of a surface at a point. Note: not a
geometric property of the surface, but parametrization dependent.
Valency A vertex in a polygonal mesh has valency niin edges emanate from the
vertex. Also: valence.
Variation diminution Intuitively: a curve or surface scheme has this property if
its output "wiggles less" than the data from which it is constructed.
Vector A direction. Usually the difference of two points.
Volume deformation A surface or a collection of surfaces may be embedded
in a cube. That cube may then be deformed using some trivariate Bezier or
B-spline method—this is the volume distortion—in order to change the shape
of the initial surface(s).
Voronoi diagram Same as Dirichlet tessellation.
Weight Rational curves and surfaces are often defined in terms of homogeneous
coordinates. The last component of the homogeneous coordinate is called weight.
Weight point The point formed by a weighted average of two control points of
a rational curve; the weights in the average are the weights of the control points.
List of Programs
The following list contains all programs that are contained in the text.
aitken: Section 7.12
area: Section 23.4
bessel_ends: Section 9.9
bez_to_points: Section 4.5
bspl_to_points: Section 8.10
curvature_0: Section 23.4
curvatures: Section 23.4
deboor: Section 8.10
deboor_blossom: Section 8.10
decas:
Section 4.5
degree_elevate: Section 6.11
direct_gspline: Section 11.7
hornbez: Section 5.8
l_u_system: Section 9.9
netcoons: Section 15.10
parameters: Section 9.9
plot_bez_surfaces: Section 16.10
plot_surf:
Section 15.10
ratbez: Section 13.10
ratbspline_to_bezier: Section 13.10
ratbspl_to_bez_surf: Section 16.10
445
446
Appendix
B
List
of
Programs
rat_subdiv:
set_up_system:
solve_system:
spline_surf_int:
subdiv_rat:
Section
13.10
Section
9.9
Section
9.9
Section
16.10
Section
13.2
Notation
H
ere is the notation used in this book.
A cross product
curve derivatives v^ith respect to the current parameter
' " curve derivatives with respect to arc length
a,
fc,
a,
j6
real numbers or real-valued functions
0 zero vector, in 3D: short for (0,0,0)
a, b points or vectors (possibly in terms of barycentric coordinates)
A, B matrices
A, B matrices v^hose elements are points (hypermatrices)
b[ intermediate points in the de Casteljau algorithm
B^
univariate Bernstein polynomials of degree n
B^
bivariate Bernstein polynomials of degree n
el,
e2, e3 short for (1,0,0), (0,1,0), and (0,0,1), respectively
b[*,
...,*] blossom
E^ J-dimensional euclidean space
D^f
directional derivative of f in the direction d
Aj difference in parameter intervals (i.e., A^ =
Ui_^i — Uj)
A^ iterated forward difference operator
Hr' cubic Hermite polynomials
/ the identity matrix
P control polygon
O an affine map
447
448 Appendix C Notation
Vi operators
t^^^ n-io\d repetition of an argument t
I
|v|
I (euclidean) length of the vector v
x^ w-partial of x(w, v)
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