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Intersection of Two Surfaces

This document discusses surfaces and intersections of surfaces in 3D space. It defines common surfaces like spheres, ellipsoids, cylinders, paraboloids and cones. It then discusses intersections of surfaces, noting that the intersection of a plane and sphere is always a circle, and the intersection of a plane and ellipsoid is always an ellipse. Examples are given of finding the projection into the x-y plane of intersections between surfaces. Appendices provide links to online resources about surfaces, integrals and theorems related to vector calculus.

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107 views20 pages

Intersection of Two Surfaces

This document discusses surfaces and intersections of surfaces in 3D space. It defines common surfaces like spheres, ellipsoids, cylinders, paraboloids and cones. It then discusses intersections of surfaces, noting that the intersection of a plane and sphere is always a circle, and the intersection of a plane and ellipsoid is always an ellipse. Examples are given of finding the projection into the x-y plane of intersections between surfaces. Appendices provide links to online resources about surfaces, integrals and theorems related to vector calculus.

Uploaded by

lzyabc597
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Surfaces and intersection of two surfaces

(A) Surfaces

sphere x 2 + y 2 + z 2 =
r2

sphere (x a ) 2 + ( y b) 2 + ( z c) 2 =
r2

x 2 + y 2 + z 2 2ax 2by 2cz + a 2 + b 2 + c 2 =


r2

x y z
ellipsoid + + =
r2
a b c

x2 y 2
2
+
=
r
a 2 b2

x2 + y 2 =
r2

elliptic cylinder

circular cylinder

y-axis

x2 + z 2 =
R2

x-axis

y +z =
r
2

circular paraboloid
=
z x +y
2

elliptic paraboloid

=
z 4x2 + y2
hyperbolic paraboloid (saddle)
=
z x y
2

Tutorial 8 Q6

z =4 x y
2

Inverted paraboloid

y2
elliptic cone x +
=
z2
4
2

B) Intersection of two surfaces

Z =4

=
z x2 + y 2

C is the intersection of the circular cylinder


x^2+y^2 = 4 and the plane x+z = 3

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

Surface defined
on the base
of cylinder

surface

Tutorial 8 Q6

Intersection of two cylinders

Volume of the solid

= 8 times of the volume of this solid


Tutorial 8 Q2

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

Intersection of two cylinders

animation

Intersection of two surfaces

The sphere x2 + y2 + z2 = 6
and the paraboloid x2 + y2 = z.

The paraboloid z = 2 x^2 y^2


and the conic surface

cone z =

x 2 + y 2 and cylinder x 2 + y 2 = 4

the sphere
and the cone.

Example 1
Determine projection into x-y plane of the curve of
intersection of a plane and a sphere

x+ y+z =
94

x2 + y 2 + z 2 =
4506

Solve the above, get rid of z, we get

x + y + (94 x y ) =
4506
2

projection into x-y plane

which is an ellipse

Example 2
Determine projection into x-y plane of the curve
of intersection of the following surfaces

z= 1 y

=
z x2 + y 2
Solve the above, get rid of z, we get

1 y = x + y
2

so

projection into x-y plane

x + 2y =
1, which is an ellipse
2

Example 3
Determine projection into x-y plane of the curve
of intersection of the following surfaces

=
z 2x2 + 3 y 2
z =
5 3x 2 y
2

Solve the above, get rid of z, we get

5x2 + 5 y 2 =
5

projection into x-y plane

1, which is a circle
x2 + y 2 =

Appendix
http://www.mhhe.com/math/calc/smithminto
n2e/cd/folder_structure/text/chap14/section0
4.htm

Greens theorem

http://www.mhhe.com/math/calc/smithminto
n2e/cd/folder_structure/text/chap10/section0
6.htm

Surfaces in space

http://www.mhhe.com/math/calc/smithminto
n2e/cd/folder_structure/text/chap14/section0
3.htm

Indep of path

http://www.mhhe.com/math/calc/smithminto
n2e/cd/folder_structure/text/chap14/section0
8.htm

Stokes Theorem

http://www.mhhe.com/math/calc/smithminto
n2e/cd/folder_structure/text/chap14/section0
6.htm

Surface integral

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