Curves and Fractals
B-Spline and Bezier
By, Amol Sawant
Amol Sawant
B-Spline Curve
• In Interpolation technique to generate a smooth
curve it must pass through all sample points.
• It means the sum of blending function must be 1.
• For every sample point for integer values of ‘u’
we may get blending function as 1, but for
fractional values of ‘u’, we may not able to pass
the curve through the sample points.
Amol Sawant
B-Spline Curve
• Suppose there are 5 sample points
• If blending function is 1 for all values of ‘u’.
Amol Sawant
• Set of Blending function which takes this the
approach of getting smooth curve by not
passing the curve through sample points is
called B-Spline.
Amol Sawant
• Basically Spline means a strip, which move
around the sample points.
• Generally B-Spline Curves are designed to
eliminate the corners of the curve.
• But to get the corners we can produce many
identical points.
Amol Sawant
Bezier Curves
• It is different way of specifying the curve.
• Same shape can be represented by B-Spline
and Bezier curve.
• Cubic Bezier curve requires the four sample
points, these points completely specify the
curve.
Amol Sawant
Bezier Curves
• If we need 2 Bezier curve connected to each
other then with 6 sample points we can
achieve it.
Amol Sawant
Bezier Curves
• The equations of the Bezier curve are.
• Here as the value of ‘a’ move from ‘o’ to ‘1’ ,
the curve travel from first point to the fourth
point.
Amol Sawant
Bezier Curves
• We can generate the Bezier curve without
referencing the above expression.
• It can be generated by simply taking
midpoints.
Amol Sawant
Cubic Bezier Curves
• To join multiple curves into one curve
smoothly, we need to specify positions and
their tangent vectors of both ends i.e.
continuity requirements.
Amol Sawant
Cubic Bezier Curves
• Four points p0,p1,p2,p3 define cubic Bezier
curve in plane or three dimensional space.
• P1 and p2 defines the tangent end points of
the curve and curve doesn’t pass from p1 and
p2
• These points are there to provide for
directional information.
Amol Sawant
Properties of Cubic Bezier Curve
• The basic functions are real in nature.
• Bezier curve always passed through the first and last
control points.
• The curve has the same end points as the guiding
polygon.
• The degree of polynomial defining the curve segment is
one less than the number of defining polygon points.
• The degree of polynomial is 3 i.e. cubic polynomial.
• The curve generally follows the shape of the defining
polygon.
• The curve lies within the convex Hull formed by the 4
sample points. Amol Sawant
Thankyou
Amol Sawant

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Curves and fractals b spline and bezier

  • 1. Curves and Fractals B-Spline and Bezier By, Amol Sawant Amol Sawant
  • 2. B-Spline Curve • In Interpolation technique to generate a smooth curve it must pass through all sample points. • It means the sum of blending function must be 1. • For every sample point for integer values of ‘u’ we may get blending function as 1, but for fractional values of ‘u’, we may not able to pass the curve through the sample points. Amol Sawant
  • 3. B-Spline Curve • Suppose there are 5 sample points • If blending function is 1 for all values of ‘u’. Amol Sawant
  • 4. • Set of Blending function which takes this the approach of getting smooth curve by not passing the curve through sample points is called B-Spline. Amol Sawant
  • 5. • Basically Spline means a strip, which move around the sample points. • Generally B-Spline Curves are designed to eliminate the corners of the curve. • But to get the corners we can produce many identical points. Amol Sawant
  • 6. Bezier Curves • It is different way of specifying the curve. • Same shape can be represented by B-Spline and Bezier curve. • Cubic Bezier curve requires the four sample points, these points completely specify the curve. Amol Sawant
  • 7. Bezier Curves • If we need 2 Bezier curve connected to each other then with 6 sample points we can achieve it. Amol Sawant
  • 8. Bezier Curves • The equations of the Bezier curve are. • Here as the value of ‘a’ move from ‘o’ to ‘1’ , the curve travel from first point to the fourth point. Amol Sawant
  • 9. Bezier Curves • We can generate the Bezier curve without referencing the above expression. • It can be generated by simply taking midpoints. Amol Sawant
  • 10. Cubic Bezier Curves • To join multiple curves into one curve smoothly, we need to specify positions and their tangent vectors of both ends i.e. continuity requirements. Amol Sawant
  • 11. Cubic Bezier Curves • Four points p0,p1,p2,p3 define cubic Bezier curve in plane or three dimensional space. • P1 and p2 defines the tangent end points of the curve and curve doesn’t pass from p1 and p2 • These points are there to provide for directional information. Amol Sawant
  • 12. Properties of Cubic Bezier Curve • The basic functions are real in nature. • Bezier curve always passed through the first and last control points. • The curve has the same end points as the guiding polygon. • The degree of polynomial defining the curve segment is one less than the number of defining polygon points. • The degree of polynomial is 3 i.e. cubic polynomial. • The curve generally follows the shape of the defining polygon. • The curve lies within the convex Hull formed by the 4 sample points. Amol Sawant