Chapter 4 Types of Surfaces 41. Imaginary points, lines, and planes Equation has no real solution The values of some of the quantities x,y,z satisfying the given conditions are imaginary. The line joining two imaginary points is real if it also contains two real points

The advantage of such statement: Many theorems and results can be stated in more general form. The formulas will be applied to imaginary elements as well as to real ones. For e.g., any one variable polynomial of degree n has n roots.

Loci of equations. The locus defined by a single equation among the variables x,y,z is called a surface. A point lies on the surface F=0 if, and only if, the coordinates of P satisfy the equation of the surface. E.g., plane, sphere The locus of the real points on a surface may be composed of curves and points, no real points

Properties of equation for a surface E.g., Properties of equation for a surface Multiplied by a constant different from zero FG=0, if and only if F=0 or G=0 F1=0,F2=0,…Fn=0, points satisfy all the n equations. Notice that the locus may be imaginary.

43. Cylindrical surface If the equation of a surface involves only two of the coordinates x,y,z, the surface is a cylindrical surface whose generating lines are parallel to the axis whose coordinate does not appear in the equation. a plane curve with generating lines perpendicular to the plane. (hint: transformation)

44. Projecting cylinders. A cylinder whose elements are perpendicular to a given plane and intersect a given curve is called: The projecting cylinder of the given curve on the given plane. Z The given plane X Y

The equation of the projecting cylinder of the curve of intersection of two surfaces F(x,y,z)=0, f(x,y,z)=0 on the plane z=0 is independent of z. The cylinder may be obtained by eliminating z between the equation of the curve.

Sylvester’s method for elimination Since the coordinates of points on the curve satisfy F=0 and f=0, they satisfy: Consider as linear equation in variables of and eliminate z and its powers, we obtain R(x,y)=0.

45. Plane sections of surfaces Intersections of a surface with planes E.g., F(x,y,z)=0, with z=k Congruent plane sections (hint: when set z=k, in the equation, the equation becomes h(x,y)=0 without z coordinates

46. Cones A surface such that the line joining an arbitrary point on the surface to a fixed point lies entirely on the surface Theorem. If the equation of a surface is homogeneous in x,y,z, the surface is a cone with vertex at the origin. P C

Proof. Let f(x,y,z) be homogeneous of degree n in (x,y,z), and let Proof. Let f(x,y,z) be homogeneous of degree n in (x,y,z), and let be an arbitrary point on the surface, so that f=0. f(0,0,0)=0, The coordinates of any point P on the line joining to the origin are , thus

Exercises P 49, No. 2, 4 47. Surface of revolution. The surface generated by revolving a plane curve about a line in its plane The fixed line is called axis of revolution. Every point of revolving curve describes a circle, whose plane is perpendicular to the axis of revolution.

Let the plane curve in z=0 be f(x,y)=0, the revolution axis be X-axis. Let , be any point on the curve, so that . Let P=(x,y,z) be any point on the circle described by . The coordinates of the center C of the circle are ; and radius is

O X Y Z

Exercises P51, No. 1,4,6