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Parametric Equation for a Line

New way of thinking about lines which will enable us to calculate the intermediate geometric transformations

• Given a line segment from v1 to v2, how to express midpoint?

  • x = 1/2 (x1 + x2) = 1/2x1 + 1/2x2

  • y = 1/2 (y1 + y2) = 1/2y1 + 1/2y2

• Three-fourths of the way?

  • x = = 1/4x1 + 3/4x2

  • y = = 1/4y1 + 3/4y2

• More generally, for a parameter t (e.g.,time)

  • x = (1 - t) x1 + tx2

  • y = (1 - t) y1 + ty2

• vt = (1 - t)v1 + tv2 called convex linear combination of v1 and v2; a linearly weighted sum
  
  

Animation - 10 MAY 95

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