lecture6: quadraticsurfaces - iu

Lecture 6: Quadratic surfaces

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

September 23, 2010

Announcements

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

I Office hours change:I Monday 10–11AM, 2–4PM, Mathematics 614.I No office hours on Wednesday.

I New TA: Sherin George <[email protected]>.Office hours: F 2–4PM in Barnard Math Help Room (Milbank 333).

I Check your e-mail.I Today’s lecture is interactive. Screenshots will be posted afterwards.

I Introduction

Conic sections review

Quadratic surfaces

Last word on lines and planes

Quadratic surfaces

A quadratic surface is a surface in space defined by a quadratic equation:{(x , y , z) | x2 + y2 = 1} Cylinder

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

{(x , y , z) | x2 + 2xy + y2 + z2 − 2z = 5} ??We study them for several reasons.

I Build 3-dimensional intuition.I Techniques useful for contour plots, which you will see more.I These surfaces are useful.I Will see some of them later in the course.

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

Quadratic surfaces

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

Quadratic surfaces

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

Quadratic surfaces

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

Introduction

I Conic sections review

Quadratic surfaces

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:I Ellipse (including circle)I HyperbolaI Parabola

{(x , y) | x2 +y2

Conic sections

{(x , y) | −x2 +y2

Conic sections

{(x , y) | y = x2}

Classification of conic sectionsTheoremThe type of a conic section

{(x , y) | Ax2 + Bxy + Cy2 + Dx + Ey + F = 0}depends on B2 − 4AC:

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

ExamplesI {(x , y) | x2 + y2 = 1}: B2 − 4AC = −4: CircleI {(x , y) | −x2 + y2 = 1}: B2 − 4AC = 4: HyperbolaI {(x , y) | x2 + 2xy + y2 + x − y = 0} = {(x , y) | (x + y)2 + (x − y) = 0}:

B2 − 4AC = 0: Parabola

Introduction

I Quadratic surfaces

Trace method

Suppose you didn’t know what {(x , y , z) | x2 + y2 + z2 = 1} represented.How could you figure it out?Fix z-coordinate to fixed value. Consider resulting curve.What happens to the curve as z varies?

Interactive graphics courtesy of Sage (http://sagemath.org),a free, open-source, and excellent mathematics software system.Available for Linux, Mac OS X, and Windows, or use it online.

3D graphics by K3DSurf (http://k3dsurf.sourceforge.net).Available for Linux, Mac OS X, and Windows.

Basic types of quadratic surfaces

Let’s try some more surfaces.I {(x , y , z) | x2 + y2/2+ z2/3 = 1}: An ellipsoidI {(x , y , z) | x2 + y2 − z2 = 1}: A hyperboloid of one sheetI {(x , y , z) | x2 + y2 − z2 = −1}: A hyperboloid of two sheetsI {(x , y , z) | x2 + y2 − z2 = 0}: A coneI {(x , y , z) | z = x2 + y2}: An (elliptic) paraboloidI {(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 differentinformation.

Let’s get some examples from you!

Introduction

Quadratic surfaces

I Uses

Parabolic reflectors

Photo by Steve Jurvetson.

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

Parabolic reflectors, cont.

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

Parabolic reflectors, cont.2

. . . or solar cooking.

Hyperboloid

By Flickr user Mélisande.

A hyperboloid can be made out ofstraight lines.

Hyperboloid gears

Taiwan’s Antique Mechanism Teaching Models Digital Museum.Model NTUT-F02 Hyperboloid Gear Mechanism

This makes hyperboloids the right shape for certain gears, when you want to changethe angle of rotation.

Hyperboloid gears in practice

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

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

Introduction

Quadratic surfaces

I Last word on lines and planes

Intersections of planes

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 normal vectors.Find it using cross product.

If you’re 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 difference.

x − y = 3 ~n2 = (1,−1, 0)~n1 × ~n2 = (1, 1,−2)

Distance to lines and planesFor a line in R2, distance is given bydot product with normal:

L = {~r ∈ R2 | ~n ·~r = ~n ·~r0}dist(~p, L) = comp~n(~p −~r0)

=~n · (~p −~r0)‖~n‖

For a plane in R3, distance is given bydot product with normal:

P = {~r | ~n ·~r = ~n ·~r0}dist(~p,P) = comp~n(~p −~r0)

=~n · (~p −~r0)‖~n‖

In both cases, sometimes easier to look at unit normal vector~n‖~n‖

QuestionWhat’s the distance from (5, 6, 7) to the plane through (1, 0, 0), (0, 1, 0), and (0, 0, 1)?

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

Distance = comp~n((5, 6, 7)− (1, 0, 0)) =(1, 1, 1) · (4, 6, 7)‖(1, 1, 1)‖

=17√3.

Distance to lines and planesFor a line in R2, distance is given bydot product with normal:

L = {~r ∈ R2 | ~n ·~r = ~n ·~r0}dist(~p, L) = comp~n(~p −~r0)

=~n · (~p −~r0)‖~n‖

For a plane in R3, distance is given bydot product with normal:

P = {~r | ~n ·~r = ~n ·~r0}dist(~p,P) = comp~n(~p −~r0)

=~n · (~p −~r0)‖~n‖

In both cases, sometimes easier to look at unit normal vector~n‖~n‖

QuestionWhat’s the distance from (5, 6, 7) to the plane through (1, 0, 0), (0, 1, 0), and (0, 0, 1)?

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

Distance = comp~n((5, 6, 7)− (1, 0, 0)) =(1, 1, 1) · (4, 6, 7)‖(1, 1, 1)‖

=17√3.

lecture6: quadraticsurfaces - iu

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