1. 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

2. 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.

3. 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

4. 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.

5. 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)

6. 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

7. 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.

8. 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.

9. 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

10. 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

11. 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

12. 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.

13. 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

14. O X Y Z

15. Exercises P51, No. 1,4,6