The Tangent Vector - Computer Graphics - Lecture Slides, Slides of Computer Graphics

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Computer

Graphics

Lecture 37

CURVES III

• Another way to define a space curve does not use intermediate points. It uses the tangents at each end of a curve, instead

• Every point on a curve has a straight line associated with it called the tangent line, which is related to the first derivation of the Parametric functions x(u), y(u), and z(u)

• From elementary calculus, we can compute, for example,

Equation (2)

( ) dx ( ) u^ du

dy u du

du

dy

Equation (3)



  =  z u k du

y u j d du

x u i d du

P u u d

• Or more simply as

P u^ =^ [ x^ u yu zu ]

Equation (4)

• We will still use the two end points, but instead of two intermediate points, we will use the tangent vectors at each end to supply the information we need to define a curve

• By manipulating these tangent vectors, we can control the slope at each end. The set of vectors , ,, and are called the boundary conditions

• We differentiate to obtain the x component of the tangent vector:

Equation (5)

dx^ ( ) u^ xu^ axu bxu cx du

d = = 3 2 + 2 +

Equation (1A)

x ( ) u = ax u^3 + bxu^2 + cxu + d x

• Evaluating (1A) and Equation 5 at u = 0, u = 1, yields