- Bezier curves and spline curves are parametric curves used in computer graphics and design. Bezier curves were developed in the 1960s and use control points to define quadratic, cubic, or higher order curves. Spline curves use piecewise polynomials to represent complex curves and surfaces for data interpolation and smoothing. - Both curves allow for local control of the shape and can interpolate or approximate data points. Bezier curves have properties like the first/last control points defining the endpoints, and the curve lying within the convex hull of all control points. Higher order curves are constructed by combining lower order curve sections.

1. Bezier Curve & Spline Curve
Presented By:
Md. Asif Shahariar
Shovan Banik
Iftekher Aziz
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2. • Parametric curve
• Known form 1912
• Pierre Bezier
• Renault Cars
• Study developed 1959 by
Paul de Castelju
Bezier Curve

3. A Bézier curve is defined by its order
(quadratic, cubic, etc.) and a set of
control points PO through P1 , the
number n of which depends on the order
(n = 2 for linear, 3 for quadratic, etc.).
The first and last control points are
always the end points of the curve;
however, the intermediate control points
(if any) generally do not lie on the
curve.
What is Bezier Curve?

4. Spline Curve
• Computer aided design & Computer Graphics
• Frequently refers to a piecewise polynomial
• Data interpolation and/or data smoothing

5. A spline curve is a mathematical
representation for which it is easy to build
an interface that will allow a user to design
and control the shape of
complex curves and surfaces.
What is Spline Curve?

6. Bezier Curve

7. Properties of Bezier Curve
• The first and last control points are interpolated.
• The tangent to the curve at the first control point is along the line joining
the first and second control points.
• The tangent at the last control point is along the line joining the second last
and last control points
• The curve lies entirely within the convex hull of its
control points.
• The Bernstein polynomials (the basis functions) sum to 1
and are everywhere positive.
• They can be rendered in many ways.

8. Properties of Spline Curve
• It provide local control of the curve shape.
• It also provide the ability to add control points without increasing the
degree of the curve
• It have the ability to interpolate or approximate a set of given data points.
• The form of a spline curve is very similar to that of a Bezier curve.

9. Bezier Curve Construction
Linear curves:
The t in the function for a linear Bézier curve can be
thought of as describing how far B(t) is from P to
P . For example when t=0.25, B(t) is one quarter of
the way from point P0 to P1 . As t varies from 0 to 1,
B(t) describes a straight line from P to P .

10. Bezier Curve Construction Con.
Quadratic curves:
For quadratic Bézier curves one can construct intermediate
points Q0 and Q1 such that as t varies from 0 to 1.
Point Q0 varies from P0 to P1 and describes a linear Bézier curve.
Point Q1 varies from P1 to P2 and describes a linear Bézier
curve.
oint B(t) varies from Q0 to Q1 and describes a quadratic
Bézier curve.

11. Bezier Curve Construction Con.
Higher-Order curves
Fourth order curves

12. Bezier Curve Construction Con.
Fifth order curves

13. Bezier Curve Application
• Bezier curves are used in the time
domain,
particularly in animation and interface
design,
e.g., a Bezier curve can be used to
specify
the velocity over time of an object such as
an
icon moving from A to B, rather than
simply
moving at a fixed number of pixels per step.

14. Bezier Curve Application Con.
• Bezier curves are widely used in computer
graphics to model smooth curves.
• Quadratic and cubic Bézier curves are most
common; higher degree curves are more
expensive to evaluate. When more complex shapes
are needed, low order Bézier curves are patched
together.

15. Design Techniques Bezier Curve
(Weights)
• Multiple control points at a
single coordinate position
gives more weight to that
position.

16. Bezier Curve Design Techniques
Con.
• When Complicated curves are to be
generated, they can be formed by piecing
several Bezier sections of lower degree
together.
• Piecing together smaller sections gives us
better control over the shape of the shape of
the curve in small region.
• It is easy to match curve section

17. Bezier Curve Design Techniques
Con.
• Since the tangent to the curve at an endpoint
is along the line joining that endpoint to the
adjacent control point.

18. Thank You