06 surfaces

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Lecture 6: Quadratic surfaces

http://www.math.columbia.edu/~dpt/F10/CalcIII/

September 23, 2010
http://www.math.columbia.edu/~dpt/F10/CalcIII/http://www.math.columbia.edu/~dpt/F10/CalcIII/


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Announcements

Midterm on Thursday, September 30. Review on Tuesday. You are allowed one handwritten page of notes, both sides. No other aids. Professor Lipshitz will administer. If you have a disability requiring accommodation, contact ODS. Do that no

Office hours change: Monday 1011AM, 24PM, Mathematics 614. No office hours on Wednesday.

New TA: Sherin George .Office hours: F 24PM in Barnard Math Help Room (Milbank 333).

Check your e-mail.

Todays lecture is interactive. Screenshots will be posted afterwards.
http://%[email protected]%3E/http://%[email protected]%3E/http://%[email protected]%3E/


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Introduction

Conic sections review

Quadratic surfaces

Uses

Last word on lines and planes


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Quadratic surfaces

A quadratic surface is a surface in space defined by a quadratic equation:

{(x,y, z)

|x2 + y2 = 1

}Cylinde

{(x,y, z) | x2 + y2 + z2 = 1} Sphere{(x,y, z) | x2 + 2xy + y2 + z2 2z = 5} ??

We study them for several reasons.

Build 3-dimensional intuition.

Techniques useful for contour plots, which you will see more. These surfaces are useful.

Will see some of them later in the course.

Basic technique: traces. Fix (say) z-coordinate to (say) 0. Consider resultResult is a quadratic curve, a conic section.


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Quadratic surfaces


{(x,y, z)

|x2 + y2 = 1

}Cylinde




Techniques useful for contour plots, which you will see more.

These surfaces are useful.




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Quadratic surfaces


{(x,y, z)

|x2 + y2 = 1

}Cylinde









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Quadratic surfaces


{(x,y, z)

|x2 + y2 = 1

}Cylinde









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Introduction


Quadratic surfaces

Uses



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Conic sections

A conic section (or quadratic curve) is defined by a quadratic equation:

{(x,y) | Ax2 + Bxy + Cy2 + Dx + Ey + F = 0}

Three basic types:

Ellipse (including circle)

Hyperbola

Parabola

{(x,y) | x2 + y2

2


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Conic sections


{(x,y) | Ax2 + Bxy + Cy2 + Dx + Ey + F = 0}

Three basic types:


Hyperbola

Parabola

{(x,y) | x2 + y2

2


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Conic sections


{(x,y) | Ax2 + Bxy + Cy2 + Dx + Ey + F = 0}

Three basic types:


Hyperbola

Parabola

{(x,y) | y = x2


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Classification of conic sections

TheoremThe type of a conic section

{(x,y)

|Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

}depends on B2 4AC: B2 4AC< 0: An ellipse (or circle, empty, or degenerate) B2 4AC = 0: A parabola (or degenerate) B2 4AC> 0: A hyperbola (or degenerate)

Examples

{(x,y) | x2 + y2 = 1}: B2 4AC = 4: Circle {(x,y) | x2 + y2 = 1}: B2 4AC = 4: Hyperbola {(x,y) | x2 + 2xy + y2 + x y = 0} = {(x,y) | (x + y)2 + (x y)

B2

4AC = 0: Parabola


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{(x,y)

|Ax2 + Bxy + Cy2 + Dx + Ey + F = 0


Examples


B2

4AC = 0: Parabola


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{(x,y)

|Ax2 + Bxy + Cy2 + Dx + Ey + F = 0


Examples


B2

4AC = 0: Parabola


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{(x,y)

|Ax2 + Bxy + Cy2 + Dx + Ey + F = 0


Examples


B2

4AC = 0: Parabola

f


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Introduction


Quadratic surfaces

Uses



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B i f d i f


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Basic types of quadratic surfaces

Lets try some more surfaces.

{(x,y, z)

|x2 + y2/2 + z2/3 = 1

}: An ellipsoid

{(x,y, z) | x2 + y2 z2 = 1}: A hyperboloid of one sheet {(x,y, z) | x2 + y2 z2 = 1}: A hyperboloid of two sheets {(x,y, z) | x2 + y2 z2 = 0}: A cone {(x,y, z) | z = x2 + y2}: An (elliptic) paraboloid

{(x,y, z) | z = x2

y2

}: A hyperbolic paraboloidYou can take traces by setting x or y to be a constant as well; that gives information.

Lets get some examples from you!

B i f d i f


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{(x,y, z)

|x2 + y2/2 + z2/3 = 1

}: An ellipsoid


{(x,y, z) | z = x2

y2



B i f d i f


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{(x,y, z)

|x2 + y2/2 + z2/3 = 1

}: An ellipsoid


{(x,y, z) | z = x2

y2



B i t f d ti f


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{(x,y, z)

|x2 + y2/2 + z2/3 = 1

}: An ellipsoid


{(x,y, z) | z = x2

y2



B i t f d ti f


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{(x,y, z)

|x2 + y2/2 + z2/3 = 1

}: An ellipsoid


{(x,y, z) | z = x2

y2



B i t f d ti f


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{(x,y, z)

|x2 + y2/2 + z2/3 = 1

}: An ellipsoid


{(x,y, z) | z = x2

y2





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{(x,y, z)

|x2 + y2/2 + z2/3 = 1

}: An ellipsoid


{(x,y, z)

|z = x2

y2

}: A hyperbolic paraboloid

You can take traces by setting x or y to be a constant as well; that gives information.




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Introduction


Quadratic surfaces

Uses


Parabolic reflectors


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Parabolic reflectors

Photo by Steve Jurvetso

A paraboloid turns out to be the ideal shape for a satellite dish. . .

Parabolic reflectors cont


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Parabolic reflectors, cont.

. . . a reflector for photography (or in a flashlight). . .

Parabolic reflectors cont 2


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Parabolic reflectors, cont.

. . . or solar cooking.

Hyperboloid


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Hyperboloid

By Flickr user Mlisande.

A hyperboloid can be madstraight lines.

Hyperboloid gears


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Hyperboloid gears

Taiwans Antique Mechanism Teaching Models Digital Museum.

Model NTUT-F02 Hyperboloid Gear Mechanism

This makes hyperboloids the right shape for certain gears, when you want

the angle of rotation.

Hyperboloid gears in practice


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Hyperboloid gears in practice

http://en.wikipedia.org/wiki/File:Differentialgetriebe2.jpg

A cut-away view of the differential in a Porsche Cayennne.

http://en.wikipedia.org/wiki/File:Differentialgetriebe2.jpghttp://en.wikipedia.org/wiki/File:Differentialgetriebe2.jpg


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Introduction


Quadratic surfaces

Uses


Intersections of planes


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The intersection of two planes is a line.With lines, main problem is to find direction vector, parallel to line.

The direction vector lies in both planes, so is perpendicular to both normaFind it using cross product.

If youre given two equations, can think of each as a plane:

x + y + z = 1 n1 = (1, 1, 1)

x

y = 3 n2 = (1,

1, 0)

n1 n2 = (1, 1,2)Also need to find one point r0 on the line. Any one solution will do.

Alternative approach: Find any two points on the line and take the differe



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x + y + z = 1 n1 = (1, 1, 1)

x

y = 3 n2 = (1,

1, 0)





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x + y + z = 1 n1 = (1, 1, 1)

x

y = 3 n2 = (1,

1, 0)





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p




x + y + z = 1 n1 = (1, 1, 1)

x

y = 3 n2 = (1,

1, 0)



Distance to lines and planes


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pFor a line in R2, distance is given bydot product with normal:

L = {r R2 | n r = n r0}dist(p, L) = compn(p

r0)

=n (pr0)

n

For a plane in R3, distancedot product with normal:

P = {r | n rdist(p,P) = compn(p

=n (pr

n

In both cases, sometimes easier to look at unit normal vectorn

n .

QuestionWhats the distance from (5, 6, 7) to the plane through (1, 0, 0), (0, 1, 0),

AnswerWe computed the normal vector earlier: n = (1, 1, 1).

Distance = compn((5, 6, 7)

(1, 0, 0)) =

(1, 1, 1) (4, 6, 7)

(1, 1, 1)

=

173

.

Distance to lines and planes


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pFor a line in R2, distance is given bydot product with normal:

L = {r R2 | n r = n r0}dist(p, L) = compn(p

r0)

=n (pr0)

n

For a plane in R3, distancedot product with normal:

P = {r | n rdist(p,P) = compn(p

=n (pr

n

In both cases, sometimes easier to look at unit normal vectorn

n .

QuestionWhats the distance from (5, 6, 7) to the plane through (1, 0, 0), (0, 1, 0),

AnswerWe computed the normal vector earlier: n = (1, 1, 1).

Distance = compn((5, 6, 7)

(1, 0, 0)) =

(1, 1, 1) (4, 6, 7)

(1, 1, 1)

=

173

.

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