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Lecture 6

Behind us

• Lines
• Planes

Coming up

Today: cylinders and quadric surfaces

Next: vector functions and curves

Read Sections 12.6, 10.1, and 13.1 of the book. We cannot cover everything in lecture or section! Some fun is left for you to have.

Questions!

Vague question

How can we relate equations to the shapes of their zero loci?

A menagerie of shapes

Cone: $x^2+y^2=z^2$

A menagerie of shapes

Freaky cylinder : $y^2=x^2(x-1)$

A menagerie of shapes

Ellipsoid: $\frac{1}{2}x^2+\frac{1}{3}y^2+z^2=1$

A menagerie of shapes

Hyperbolic paraboloid: $\frac{1}{9}x^2-\frac{1}{4}y^2=z$

A menagerie of shapes

Elliptic paraboloid: $\frac{1}{9}x^2+\frac{1}{4}y^2=z$

Key idea

• Slice the shape with planes and reassemble the pieces.
• This idea recurs throughout the study of geometry (even by professionals!).
• Subdividing, solving, reassembling is also how computers graph these things.

Example: $\frac{1}{9}x^2+\frac{1}{4}y^2=z$

We can make a horizontal trace (horizontal slice) at $z=6$.

Example: $\frac{1}{9}x^2+\frac{1}{4}y^2=z$

We can make a horizontal trace (horizontal slice) at $z=6$.

Example: $\frac{1}{9}x^2+\frac{1}{4}y^2=z$

We can make a horizontal trace (horizontal slice) at $z=6$.

• Equation for the curve in the horizontal plane: $$\frac{1}{9}x^2+\frac{1}{4}y^2=6.$$
• What shape is this?
• What shape will a general horizontal trace have?

Example: $\frac{1}{9}x^2+\frac{1}{4}y^2=z$

We can also make a vertical trace (vertical slice) at $y=0$.

Example: $\frac{1}{9}x^2+\frac{1}{4}y^2=z$

We can also make a vertical trace (vertical slice) at $y=0$.

Example: $\frac{1}{9}x^2+\frac{1}{4}y^2=z$

We can make a vertical trace (vertical slice) at $y=0$.

• Equation for the curve in the vertical plane: $$\frac{1}{9}x^2=z.$$
• What shape is this?
• What shape will a general vertical trace have?

Sketching the shape

• Choose some horizontal traces and sketch them
• Choose some vertical traces and sketch them
• Hope for the best

Demonstration

Let's try the equation $x^2+y-z=0$.

• General horizontal trace: $y=-x^2+a$, $a$ a constant.
• General $xz$-plane vertical trace: $z=x^2+b$, $b$ a constant.
• General $yz$-plane vertical trace: $z=y+c$, $c$ a constant.
• Draw some!

Demonstration

Final assembled product

Cosmic taco: $x^2+y-z=0$.

Practice

• Calculate horizontal and vertical traces for the hyperbolic paraboloid: $\frac{1}{9}x^2-\frac{1}{4}y^2=z$
• How do the traces help you to sketch the object?
• How do the traces help you to tell objects apart?

New shapes to consider

Apply the techniques we've been discussing to to draw sketches of the solutions to these equations in three variables.

• $x^2+y^2-z^2=1$
• $z^2-x^2-y^2=1$
• $x^2+y^2=1$
• $x^2-2x+y^2-z^2=0$

Next time: vector functions and curves!