Intersection of Two Surfaces

2 2 2 2 sphere x y z r 2 2 2 2 ( ) ( ) ( )sphere x a y b z c r

2 2 2

2 x y z

ellipsoid ra b c

Surfaces and intersection of two surfaces

(A) Surfaces

2 2 2 2 2 2 2 2 2 2x y z ax by cz a b c r

2 2 2x y r

circular cylinder

2 2

2

2 2

x yr

a b

elliptic cylinder

2 2 2x z R

y-axis

x-axis 2 2 2y z r

2 2

circular paraboloid

z x y 2 2

4

elliptic paraboloid

z x y

2 2

(saddle)hyperbolic paraboloid

z x y

Tutorial 8 Q6

Inverted paraboloid 2 24z x y

2

2 2 4

yelliptic cone x z

2 2z x y

4Z

C is the intersection of the circular cylinder x^2+y^2 = 4 and the plane x+z = 3

(B) Intersection of two surfaces

Intersection of plane and sphere is always circle

Intersection of plane and ellipsoid is always ellipse

Intersection of plane and cylinder

Intersection of plane and paraboloid

Intersection of hyperbolic paraboloid (saddle) and cylinder

Tutorial 8 Q6 Surface defined on the base of cylinder

surface

http://www.math.tamu.edu/~Tom.Kiffe/calc3/newcylinder/2cylinder.html

Intersection of two cylinders

Volume of the solid

8 times of the volume of this solid

Tutorial 8 Q2



Intersection of two cylinders

animation

The sphere x2 + y2 + z2 = 6

and the paraboloid x2 + y2 = z.

The paraboloid z = 2 − x^2 − y^2 and the conic surface

Intersection of two surfaces

2 2 2 2 cone and cylinder 4z x y x y

the sphere and the cone.

94x y z 2 2 2 4506x y z

Example 1

Determine projection into x-y plane of the curve of intersection of a plane and a sphere

Solve the above, get rid of z, we get

2 2 2(94 ) 4506x y x y

which is an ellipse

projection into x-y plane

Example 2

Determine projection into x-y plane of the curve of intersection of the following surfaces

21z y

2 2z x y

Solve the above, get rid of z, we get

2 2 21 y x y

so 2 22 1, which is an ellipsex y


Note: Parametric equation of the curve of intersection is

21 1cos , sin , 1 sin

22x y z

or 21 1( ) cos i+ sin j+(1- sin )k

22r

Example 3

Determine projection into x-y plane of the curve of intersection of the following surfaces

2 22 3z x y

2 25 3 2z x y Solve the above, get rid of z, we get

2 25 5 5x y

2 2 1, which is a circlex y


Note: Parametric equation of the curve of intersection is 2 2cos , sin , 2cos 3sinx y z

2 2( ) cos i+sin j+(2cos 3sin )kr

http://www.mhhe.com/math/calc/smithminton2e/cd/folder_structure/text/chap14/section04.htm

Green’s theorem



Surfaces in space http://www.mhhe.com/math/calc/smithminton2e/cd/folder_structure/text/chap10/section06.htm


Surface integral

Stokes’ Theorem

Indep of path

Appendix
















Intersection of Two Surfaces

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