Graphics Reference
In-Depth Information
p , in which an input value of 4 would generate values of 2 or
x (such as y ¼
2). Implicit equations
are of the form f ( x , y )
0. The implicit form is good for testing to see if a point is on a curve because the
coordinates of the point can easily be put into the equation for the curve and checked to see if the equa-
tion is satisfied. The drawback of the implicit form is that generating a series of points along a curve is
often desired, and implicit forms are not generative. Parametric equations are of the form x ¼ f ( t ),
y ¼ g ( t ). For any given value of t , a point ( x , y ) is generated. This form is good for generating a sequence
of points as ordered values of t are given. The parametric form is also useful because it can be used for
multivalued functions of x , which are problematic for explicit equations.
Functions can be classified according to the terms contained in them. Functions that contain only
variables raised to a power are polynomial functions. If the highest power is one, then the function is
linear . If the highest power is two, then the function is quadratic . If the highest power is three, then it is
cubic . The highest power of a polynomial function of a single variable is referred to as the degree of the
polynomial. If the function is not a simple polynomial but rather contains sines, cosines, log, or a vari-
ety of other functions, then it is called transcendental . In computer graphics, the most commonly
encountered type of function is the cubic polynomial.
Continuity refers to how well behaved the curve is in a mathematical sense. For a value arbitrarily
close to an x 0 if the function is arbitrarily close to f ( x 0 ), then it has positional ,or zeroth-order ,
continuity ( C
¼
0 ) at that point. If the slope of the curve (or the first derivative of the function) is con-
tinuous, then the function has tangential ,or first-order ,continuity( C
1 ). This is extended to all of the
function's derivatives, although for purposes of computer animation the concern is with first-order
continuity or, possibly,
2 ). Polynomials are infinitely
second-order ,or curvature , continuity ( C
continuous.
If a curve is pieced together from individual curve segments, one can speak of piecewise
properties— the properties of the individual pieces. For example, a sequence of straight line seg-
ments, sometimes called a polyline or a wire , is piecewise linear. A major concern regarding piece-
wise curves is the continuity conditions at the junctions of the curve segments. A junction has C
0
0
continuity ,oris C
continuous , if one curve segment begins where the previous segment ends. This
is referred to as zeroth-order ,or positional , continuity at the junction. If the beginning tangent of one
curve segment is the same as the ending tangent of the previous curve segment, then there is
first-order ,or tangential , continuity at the junction. The junction is C
1 continuous if it has tangential
and positional continuity. If the beginning curvature of one curve segment is the same as the ending
curvature of the previous curve segment, then there is second-order ,or curvature , continuity at the
junction. The junction is C
2 continuous if it has curvature, tangential, and positional continuity.
Typically, computer animation is not concerned with continuity beyond second order.
Sometimes in discussions of the continuity at segment junctions, a distinction is made between
parametric continuity and geometric continuity (e.g., [ 14 ]). So far the discussion has concerned para-
metric continuity. Geometric continuity is less restrictive. First-order parametric continuity, for exam-
ple, requires that the ending tangent vector of the first segment be the same as the beginning tangent
vector of the second. First-order geometric continuity, on the other hand, requires that only the direc-
tion of the tangents be the same, and it allows the magnitudes of the tangents to be different. Similar
definitions exist for higher order geometric continuity. One distinction worth mentioning is that para-
metric continuity is sensitive to the rate at which the parameter varies relative to the length of the curve
traced out. Geometric continuity is not sensitive to this rate.
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