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Splines

Splines. By: Marina Uchenik. History and Dilemma. Early designers of ship hulls or air crafts were faced with the problem of how to create curvatures They began by bending large strips of wood or metal and tracing them with chalk

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Splines

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  1. Splines By: Marina Uchenik

  2. History and Dilemma • Early designers of ship hulls or air crafts were faced with the problem of how to create curvatures • They began by bending large strips of wood or metal and tracing them with chalk • Then facing another problem if they needed one part to be modeled in one direction while the other part needed to be in a different direction

  3. Solution • Isaac Schoenberg introduced the idea of using mathematical splines • He found that for small deflections, the shape of the physical spline was a piece-wise cubic polynomial. • The development of parametric spaces arose

  4. What is a Spline? • An approximated curve through previously defined points • Many cubic functions pieced together to create a free-form shape

  5. How do we approximate one? • Cubic Splines • Given n + 1 data points P0 , P1 , ..., Pnwe want to construct a curve which passes through them. • Gobal control: moving one of the control points affects the entire curve. • Do not require geometric constraints, such as tangent directions or control points, it can be derived as a set of scalar functions Si(x).

  6. Approximation Cont. (1) • B-Splines • Most basic method which uses cubic splines. • Only approximate the position of a set of points. • Local control: moving one data point only changes a portion of the curve. • Useful in geometric design and modeling.

  7. Approximation Cont. (2) • Hermite Curve • Defined by two end points P1 and P2 and their respective tangent directions T1 and T2.

  8. Approximation Cont. (3) The Bézier curve is given by the parametric function: • Bézier Curves • Defined for any polynomial of degree n, given n+1 control points P1 , P2 , ..., Pn n ∑ B(t) = Bn,k(t) Pk k = 0 where the blending functions Bn,k are the Bernstein polynomials defined by: Bn,k(t) = ( ) tk (i -t)(n-k) n k

  9. Approximation Cont. (4) • Uniform B-Splines • Uses parametric cubic polynomials on a uniform knot sequence of successive integers • Non-Uniform B-Splines • Most widely used • Involves the use of irregularly spaced knot sequences

  10. Who uses them? • CAD • Any type of computer aided design • Their initial problem was to be able to completely define a shape • Before splines, designers could only use existing shapes pushed together to create other shapes; but there are only so many shapes that could be constructed by using cubes, cylinders, and cones

  11. Who else uses them? • Animated and 3D films • Simulated surgery and other educational programs • Analysis of particle movement • Paths of projectiles

  12. How do they make my life better? • Aerodynamic sports cars • Traveling safely in a perfectly designed plane • Avoiding frog disections

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