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Lecture No.37 The Tangent Vector
Another way to define a space curve does not use intermediate points. It uses the tangents at each end of the 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), such as those given by Equation 2 of previous lecture. Thus
z u du
d yu and du
d xu du
d , , (1)
From elementary calculus, we can compute, for example,
dx u du
dyu du du
dy /
We can treat dx u du , dy u du , anddz u du as components of a vector along the
tangent line to the curve. We call this the tangent vector, and define it as
(^)
(^) duzuk
d yuj du
d xui du
d P u^ u (3)
Or more simply as
P u^ xu yu z^ u (4)
(Here the superscript u indicates the first derivative operation with respect to the independent variable u). This is a very powerful idea, and we will now see how to use it to define a curve.
In the last section, we discussed how to define a curve by specifying four points. Now we have another way to define a curve. 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 p 0 , p 1 , pu 0 , and p 1 u are called the
boundary conditions. This method itself is called the cubic Hermite interpolation, after C. Hermite (1822-1901) the French mathematician who made significant contributions to our understanding of cubic and quadratic polynomials.
We differentiate to obtain the x component of the tangent vector:
dx u xu^ axu bxu cx du
d 3 2 2 (5)
Figure 1 Defining a curve using end points and tangent vectors.
x u ax u^3 bxu^2 cxu dx (1A)
P 1 P 0
P u 1
P u 0 docsity.com
Evaluating (1A) and Equation 5 at u = 0, u = 1, yields
x x x
u
x
u
x x x x
x
x a b c
x c
x a b c d
x d
0
1
0
(6)
Using these four equations in four unknowns, we solve for a (^) x , bx , cx and d (^) x in terms of the boundary conditions
0
0
0 1 0 1
0 1 0 1 3 2
d x
c x
b x x x x
a x x x x
x
u x
u u x
u u x
Substituting the result into Equation (1A), yields
x^ u^ 2 x 0 2 x 1 x 0 u^ x 1 u u^3 ^ 3 x 0 3 x 1 2 x 0 u x 1 ux 0 u u^2 x 0 u x 0 (8)
Rearranging terms we can rewrite this as
x u 2 u 3 3 u^21 x 0 2 u^33 u^2 x 1 u 3 2 u^2 u x u 0^ u 3 u^2 x 1 u (9)
Because y(u) and z(u) have equivalent forms, we can include them by rewriting Equation 9 in vector form:
p u 2 u 3 3 u^21 p 0 2 u^33 u^2 p 1 u 3 2 u^2 u pu 0^ u 3 u^2 p 1 u (10)
To express Equation 10 in matrix notation, we first define a blending function matrix F F 1 F 2 F 3 F 4 , where
3 2 4
3 2 3
3 2 2
3 2 1
F u u
F u u u
F u u
F u u
These matrix elements are the polynomial coefficients of the vectors which we rewrite as
p u F 1 p 0 F 2 p 1 F 3 pu 0^ F 4 p 1 u (12)
If we assemble the vectors representing the boundary conditions into a matrix B ,
u u^ T B p 0 p 1 p 0 p 1 (13)
Then
p u FB (14)
Here again we write the matrix F as the product of two matrices, U and M, so that
F UM (15)
where
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B =
11
00
1
0
m t
mt
P
P
Carefully consider this array of 12 elements; they uniquely define the curve. By changing either m 0 or m 1 , or both, we can change the shape of the curve. But it is a restricted kind of change because not only do the end points remain fixed, but the end slopes are also unchanged!
The three curves drawn with light lines in Figure 3 show the effects of varying m0 and m1. This is a very powerful tool for designing curves, making it possible to join up end- to-end many curves in a smooth way and still exert some control over the interior shape of each individual curve. For example, as we increase the value of m 0 while holding m 1 fixed, the curve seems to be pushed toward p 1. Keeping m 0 and m 1 equal but increasing their value increases the maximum deflection of the curve from the x-axis and increases the curvature at the maximum. (Under some conditions, not necessarily desirable, we can force a loop to form).
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