BEZIER CURVES

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BEZIER CURVES

• Dr. Regalla Srinivasa Prakash

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Bezier Curves

• Approximator Adds flexibility and intuitive
  feel in mechanical design
• Credited to P. Bezier P. De Casteljau also
  developed independently
• Only control points (not necessarily only 2 per
  segment) form the input, no specification of
  tangent vectors provides more direct relation
  between input (points) and output (curve)
• The order of the Bezier is variable (unlike HCC)
  and depends on the number of control points,
  (n1)
• (n1) number of control points define an nth
  degree curve (we will term degree of curve as
  (k-1)
• Bezier curve can be smoother than HCC

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• The Bezier curve is defined in terms of (n1)
  control points, called as control points
• These control points form the vertices of the
  control or Bezier characteristic polygon
• The control or Bezier characteristic polygon
  uniquely defines the curve

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• The curve shape tends to follow the polygon shape
  and thus easy to guess and sketch
• Order of the control points is very essential
  because changing the order can change the shape
  of the curve

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Bezier Curve

• The user supplies (n1) control points, Pi
• Write the curve as

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• The functions Bi,n are the Bernstein polynomials
  of degree (k-1)n
• We can write the complete form of Bezier curve
  while observing that C(n,0)C(n,n)1

Special case, 3 control points B(u) P0 ( 1 -
u ) 2 P1 2 u ( 1 - u ) P2 u2 Special
case, 4 control points B(u) P0 ( 1 - u )3
P1 3 u ( 1 - u )2 P2 3 u2 ( 1 - u )
P3 u3
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• Substitute u0 and u1,?
• The curve is tangent to the first and last
  segments of control polygonal segmentsproof?
• Deriving the parametric form r times

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Bezier Basis Functions for (n1)4
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Some Bezier Curves
What did you observe??
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Characteristics of Bezier Curve

• 1) The curve interpolates the first and last
  control points.
• It passes through Po and Pn if we substitute u0
  and u1

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• 2) The curve is tangent to the first and last
  curve segments of the control polygon.

• Can you find the second derivative at P0? What do
  you observe?
• Ans It is determined by Po, P1 and P2. In
  general, the rth derivative at an endpoint is
  determined by its r neighbouring control points.

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• 3) The curve is symmetric with respect to u and
  (1-u).
• That is, reversing the direction of
  parametrization does not change the curve shape.
  To check this property substitute 1-u v.
• The symmetry of the Bezier curve is due to the
  symmetry (NOT equality) of the Bernstein basis,
  which in turn is due to the property of the
  binomial functions C(n, i) C(n, n-i)
• Which makes Bi,n(u) and Bn-i,n(u) to be symmetric
  when plotted with respect to u.
• Be careful, Bi,n(u) ? Bn-i,n(u), never, except
  at u0.5.

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• 4) The polynomial Bi,n has a maximum value of
  C(n,i)(i/n)i(1 i/n)n-i occuring at ui/n
• This can be obtained by setting dBi,n/du 0
• This means that each ith control point is most
  influential on the shape of the curve at ui/n,
  where (n1) are the total control points.
• For example in a cubic Bezier curve defined by
  Po, P1, P2 and P3, these points are most
  influential at u0, 1/3, 2/3 and 1.

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• 5) The shape of the curve (for a given set of
  control vertices) can be changed by two methods.
• By moving a control point
• By providing multiple control points at a vertex
  of control polygon. Multiplicity of control
  points makes the curve pulled towards that point.

Multiple control points are specified here
K3
K2
K1
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• 6) A closed Bezier curve can be obtained by
  making Po and Pn coincident. Will the degree of
  the curve change?

By making the last point co-incide with the first
point
By adding additional control point at the first
point
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• 7) Partition of Unity Property
• For any valid value of u, for any degree of given
  curve

This provides for invariance of Bezier curve
(or invariance of the relationship between the
Bezier curve and its control polygon) under
affine transformations. Method to check
numerical computation and software developments.
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• 8) Positivity
• For any value of u, given any degree of curve,
  each Bernstein polynomial satisfies

This ensures that the Bezier curve lies entirely
in its convex hull.
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The Convex Hull Property of Bezier Curve

• A curve is said to have convex hull property if
  it entirely lies in its convex hull defined by
  its control polygon.
• The Bezier curve defined with Bernstein
  polynomial basis possesses the convex hull
  property

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How to get the Convex Hull of a Bezier curve?

• The convex hull of a Bezier curve is the maximum
  area obtained by joining all control points to
  all control points.

Convex Hull (Hatched region)
Example 1 Degree 3
Example 2 Degree 3
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Convex Hull (Hatched region)
Example 3 Degree ?
Example 4 Degree ?
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Consequences of Convex Hull Property

• There are three important aspects
• Degeneration to a straight line useful
• Clipping like manipulations become easy
• Wild oscillations of the curve about any control
  point is avoided

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Repeated SlideConsequences of Convex Hull
Property

• There are three important aspects
• If the control polygon degenerates to a straight
  line, the curve segment also should become
  straight line. This is a useful design feature
  and is assured by convex hull property.
• Clipping like manipulations need to clip only the
  control polygon not the actual curve, thus saving
  in computational time.
• Wild oscillations of the curve about any control
  point is avoided.

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Composite Bezier Curves Blending

• For Co continuity the last control of the first
  Bezier curve segment be coincided with the first
  point of the second Bezier curve segment.
• For C1 continuity The end slope of the first
  segment need to be equal to the starting slope of
  the second segment. This requires that the
  (n-1)th point of the first segment, the joining
  point and the second point of the second segment
  be collinear.
• Now
• (n1) Number of the control points of the first
  segment
• (m1) Number of the control points of the
  second segment
• Equating the tangent vectors
• n(Pn Pn-1)first segment m(P1 Po)second
  segment
• (Pn Pn-1)first segment (m/n)(P1
  Po)second segment

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Example

• If the first segment is formed by Po, P1, P2 and
  P3 and second segment is formed by P4, P5, P6, P7
  and P8, then P2, P3 (coinciding with P4) and P5
  must be collinear. Therefore
• (P4 P3) (4/3)(P5 P4)

P7
P2
P4
P1
P8
P3
P5
P6
Po
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Sub-Dividing Bezier Curves de Casteljaus
Algorithm

• Step 1 Find the midpoints of the lines joining
  the original control vertices. Call them M01,
  M12, M23
• Step 2 Find the midpoints of the lines joining
  M01, M12 and M12, M23. Call them M012, M123
• Step 3 Find the midpoint of the line joining
  M012, M123. Call it M0123
• The curve with control points P0, M01, M012 and
  M0123 exactly follows the original curve from the
  point with u0 to the point with u0.5
• The curve with control points M0123 , M123 , M23
  and P3 exactly follows the original curve from
  the point with u0.5 to the point with u1

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Sub-Dividing Bezier Curves
M12
P1
P2
M012
M0123
M123
M01
M23
P0
P3
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Sub-Dividing Bezier Curves
P1
P2
P0
P3
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de Casteljaus Algorithm

• You can find the point on a Bezier curve for any
  parameter value u with a similar algorithm
• Say you want u0.25, instead of taking midpoints
  take points 0.25 of the way

M12
P2
P1
M23
t0.25
M01
P0
P3
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• Bit questions

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• Who invented Bezier curve?
• Bézier curve is attributed and named after a
  French engineer, Pierre Bézier, who used them for
  the body design of the Renault car in the 1970's.
  

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• What are the applications of Bezier curve, past
  and present?
• They have since obtained dominance in the
  typesetting industry and in particular with the
  Adobe Postscript and font products.

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• If only one control point is given, what will be
  the Bezier curve?
• If there is only one control point P0, i.e., n0,
  then P(u) P0 for all u.

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• If there are only two control points P0 and P1,
  what will be the nature of Bezier curve?
• i.e., n1, then the formula reduces to a line
  segment between the two control points.

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• Prove that the Bezier curve with Bernstein
  polynomials in general does not pass through any
  of the control points except the first and last.
• Solution
• From the formula P(0) P0 and P(1) Pn.

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• Prove that the curve is always contained within
  the convex hull of the control points and that it
  never oscillates wildly away from the control
  points.
• Solution Positivity property.

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• How do you obtain the closed Bezier curve? What
  is the effect on the degree of the curve?
• Solution Closed curves can be generated by
  making the last control point the same as the
  first control point. First order continuity can
  be achieved by ensuring the tangent between the
  first two points and the last two points are the
  same.

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• What is meant by multiplicity of control points
  and what is the consequence of it? Will multiple
  control points at a coordinate location change
  the degree of the curve?
• Solution
• Adding multiple control points at a single
  position in space will add more weight to that
  point "pulling" the Bézier curve towards it.

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• Why it is necessary to blend multiple Bezier
  curve segments to model a part represented by a
  set of data points, instead of building a single
  curve?
• Solution
• As the number of control points increases it is
  necessary to have higher order polynomials and
  possibly higher factorials. It is common
  therefore to piece together small sections of
  Bézier curves to form a longer curve. This also
  helps control local conditions, normally changing
  the position of one control point will affect the
  whole curve. Of course since the curve starts and
  ends at the first and last control point it is
  easy to physically match the sections. It is also
  possible to match the first derivative since the
  tangent at the ends is along the line between the
  two points at the end.

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• Does Bezier curve provide C2 continuity at the
  blend point?
• Solution
• Second order continuity is generally not
  possible.

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• Is it possible to build two parallel Bezier
  curves?
• Solution
• Except for the redundant cases of 2 control
  points (straight line), it is generally not
  possible to derive a Bézier curve that is
  parallel to another Bézier curve. It isn't
  possible to create a Bézier curve that is
  parallel to another, except in the trivial cases
  of coincident parallel curves or straight line
  Bézier curves.

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• Can a circle be exactly represented by a Bezier
  curve?
• Solution
• A circle cannot be exactly represented with a
  Bézier curve.

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• What are the advantages of Bezier curve? Is it
  possible to build interpolating Bezier curve?
• Solution
• Bézier curves have wide applications because they
  are easy to compute and very stable. There are
  similar formulations which are also called Bézier
  curves which behave differently, in particular it
  is possible to create a similar curve except that
  it passes through the control points.

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How the degree of Bezier curve is decided? Is the
curve ever symmetric?

• Soution
• The degree of the curve is one less than the
  number of control points, so it is a quadratic
  for 3 control points. It will always be symmetric
  for a symmetric control point arrangement.

Courtesy http//astronomy.swin.edu.au/pbourke/cu
rves/bezier/
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What is the advantage of tangency of Bezier curve
to the first and last control polygonal segments?

• The curve always passes through the end points
  and is tangent to the line between the last two
  and first two control points. This permits ready
  piecing of multiple Bézier curves together with
  first order continuity.

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What is meant by the curve well behaving?

• The curve always lies within the convex hull of
  the control points. Thus the curve is always
  "well behaved" and does not oscillating
  erratically.

Courtesy http//astronomy.swin.edu.au/pbourke/cu
rves/bezier/
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What are the limitations of Bezier Curve?

• Solution
• When interpolation is required, Bezier curve can
  not provide that feature.
• There is no local control, only global control is
  available
• When a large number of control points are to be
  modeled by a single curve segment, Bezier curve
  is practically impossible because the degree of
  the curve has to be very high, exactly one less
  than the number of control points. Moreover, it
  will only be local control over the entire
  portion the curve.

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What is invariance of Bezier curve?

• Translational invariance means that translating
  the control points and then evaluating the curve
  is the same as evaluating and then translating
  the curve
• Rotational invariance means that rotating the
  control points and then evaluating the curve is
  the same as evaluating and then rotating the
  curve
• These properties are essential for parametric
  curves used in graphics
• It is easy to prove that Bezier curves, Hermite
  curves and everything else we will study are
  translation and rotation invariant
• Some forms of curves, rational splines, are also
  perspective invariant
• Can do perspective transform of control points
  and then evaluate the curve