identifying normal is somewhat easier on surfaces than on curved surfaces
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Answer:
e have seen the simplest curves (lines) and surfaces (planes) in the previous page. Next to lines and planes, there are conics and quadric surfaces. Although conics and quadric surfaces have been around for about 2000 years, they are still the most popular objects in many computer aided design and modeling systems. We shall discuss conics, quadric surfaces and tori on this page only. Consulting your calculus and/or geometry books should be very helpful. Your linear algebra book should also cover some of these topics with a modern approach.
The following figures show to you three different ways of cutting a cone with a plane. The conic sections, from left to right, are an ellipse, a hyperbola and a parabola.
Curves
Circles
The simplest non-linear curve is unquestionably the circle. A circle with center (a,b) and radius r has an equation as follows:
(x - a)2 + (x - b)2 = r2
If the center is the origin, the above equation is simplified to
x2 + y2 = r2
The above equations are referred to as the implicit form of the circle. The parametric form of a circle is
x = rcos(t)
y = rsin(t)
The following is the parametric form of a circle whose center is not the origin:
x = a + rcos(t)
y = b + rsin(t)
The above parametric form uses trigonometric functions. We shall discuss a parametric form of a circle without trigonometric functions later.
Conics in Normal Forms
A direct generalization of the circle is the so-called conic curves or simply conics. Greeks knew about conics very well. In fact, Apollonius of Perga (262 - 200 B.C.) wrote a book of several volumes about conics. Conics are the intersection curves of a plane and a circular cone (i.e., a cone whose base is a circle and whose axis is perpendicular to the base and through the center of the base circle).
There are three types of non-degenerate conics: ellipses, hyperbolas and parabolas. Ellipses and hyperbolas are called central conics because they have a center of symmetry, while parabolas are non-central.
The axes of this ellipse are the x- and y-axis, a and b are the axis lengths, and the larger one of a and b is the major axis while the smaller one is the minor axis. It is not difficult to see that an ellipse in this form has the following parametric form:
x = acos(t)
y = bsin(t)
The normal form of a hyperbola is the following implicit equation:
The definition of major axis and minor axis are identical to that of ellipses. The x-axis intersects the curve at two points (a, 0) and (-a, 0) and, the y-axis does not intersect the curve at all. A possible parametric form of the above hyperbola is the following:
x = asec(t)
y = btan(t)
The center of an ellipse and hyperbola, in normal form, is the coordinate origin and the curve is symmetric about its center and its axes.
The normal form of a parabola is the following implicit equation:
In this normal form, for any point (x,y) on a parabola, the value of y must be positive and the opening of this parabola is upward. The axis of this parabola is the y-axis. It is intersecting to note that the normal form of a parabola is already a parametric form. Or, if you like, you can rewrite it into the following:
x = t
y = t2 / (4p)
Conics in General Form
Conics are degree two curves because their most general form is the following degree two implicit polynomial:
In the above polynomial, the coefficients of xy, x and y are 2B, 2D and 2E, respectively. This polynomial has six coefficients; however, dividing it with a non-zero coefficient would reduce six to five. Thus, in general, five conditions can uniquely determine a conic. In linear algebra, you perhaps have learned the way of reducing the above polynomial to a normal form using eigenvalues and eigenvectors.
Frequently, we only want to know the curve type of a general second degree polynomial. In this case, as long as the second degree equation represents a conic rather than two intersecting or parallel lines, it can easily be done as follows:
If B2 < A*C, the general equation represents an ellipse.
IF B2 = A*C, the general equation represents a parabola.
If B2 > A*C, the general equation represents a hyperbola.
Expression B2-A*C is called the discriminant of the general second degree polynomial. Based on the above, if the value of the discriminant is less than, equal to or greater than zero, the conic is an ellipse, a parabola, or a hyperbola.
Conics in Matrix Form
One nice thing of conics is that its general form can be rewritten compactly using matrices. First, each point x = (x, y) is considered as a column vector whose third component is 1 and hence the transpose is a row vector, written as xT = [ x, y, 1 ]. Next, the six coefficients of the general second degree polynomial are used to construct a three-by-three symmetric matrix as follows:
It is not difficult to verify that the general second degree polynomial becomes
Now what you have learned from linear algebra can be applied to this matrix form.
