Introduction to curves and surfaces

Question 1

Bézier curve: origin

Answer 1

• Developed to form curves in CAD/CAM systems for car industry
• Now Bézier curves are contained in almost all CAD systems

Question 2

Bézier curve: definition

Answer 2

• The shape of a Bézier curve is manipulated by some control points, which form the control polygon
• The curve is obtained from the control points by applying the de Casteljau algorithm:
  a parameter varies from 0 to 1;
  accordingly, on each segment of the control polygon a point is found;
  the points are linked finding a new polygon with one segment less;
  iteration until a point remains, which a point of the Bezier curve

Question 3

Degree of the Bézier curve

Answer 3

It’s equal to the number of segments of the control poly, or one less than the number of control points

Question 4

Properties of the Bézier curve

Answer 4

1) Convex hull property: a Bézier curve lies in the convex hull of its control polygon
2) Variation diminishing property in the plane: a Bézier curve does not change the side of a straight line more often than its control polygon

Question 5

Principal problem of Bézier curves and possible solution

Answer 5

A change of a single control point changes the entire curve, even if the variation far from the changed point is small.
It should be better to have a local behaviour, so spline curves are used, which are a sequence of low order Bézier curves, joined to have appropriate transiotion properties.

Question 6

Spline curve: definition

Answer 6

• Union of low degree Bézier curves, with appropriate transition conditions, for example common tangent
• Locally controllable
• The degree of the spline curve is the degree of the Bézier segments
• The control points of the spline are often different from the control points of the Bézier segments
• A particular spline curve is the B-spline, very often used in CAM systems

Question 7

B-spline curve: definition

Answer 7

A B-spline curve of degree d is a sequence of Bézier curves of degree d, put together with optimal smoothness

Question 8

B-spline curve: properties

Answer 8

• The curve lies in the convex hull of its control polygon

- variation diminishing property

Question 9

NURBS: definition

Answer 9

• Non-uniform-rational-B-splines
• extension of B-splines, with weights
• weights equal 1 - > B-spline
• increasing a weight of a control point pulls the curve near that point; increasing the weights of every control point, however, does not change the curve
• necessary to represent any type of conics

Question 10

Bézier curve: mathematical definition

Answer 10

Bézier curve of degree n (order n+1)
P(t) = sum(i=0,n) Bn,i(t)*Pi
where
Bn,i(t) are the Bernstein polynomials of order n+1
Bernstein polynomial are the Bézier blending functions

Question 11

B-spline curve: mathematical definition

Answer 11

• To solve the problem of global properties of the Bézier curve, it’s possible to use blending functions that are zeros except for some values of t

B-spline curve of degree k (order k+1):
P(t) = sum(i=0,n) Nk,i(t) * Pi
Nk,i(t) are the blending functions of order k+1 (constant, triangle, parabola)

Question 12

NURBS: Mathematical definition

Answer 12

P(t) = sum(i=0,n) Ri,k(t) * Pi

Ri,k = Ni,k(t)wi / sum(j=0,n) Nj,kwj

Question 13

Bézier surface: Mathematical definition

Answer 13

S(u,v) = sum(i=0,n)sum(j=0,m) Bn,i(u) * Bm,j(v) * Pi,j

Question 14

B-spline surface: Mathematical definition

Answer 14

S(u,v) = sum(i=0,n)sum(j=0,m) Nk,i(u) * Nq,j(v) * Pi,j

Question 15

NURBS surface: Mathematical definition

Answer 15

S(u,v) = sum(i=0,n)sum(j=0,m) Ri,j(u,v) * Pi,j

Ri,j(u,v)=…