Quadratic Surfaces

Question 1

What is the trace of a surface?

Answer 1

Curve of intersection of the surface with planes parallel to the coordinate planes

Question 2

hat

How do you sketch a quadratic surface using traces?

Answer 2

1. Identify the xy, yz, xz, planes
2. Let variable that determines position of plane = k, calculate interval in which it’s allowed to vary
3. Use conic section equations to recognise specific graphs

Question 3

What is the formula for an ellipsoid and when is it a sphere?

Answer 3

x+y+z=1
When divisions are equal

Question 4

What is the formula for a cone and when is the trace a pair of lines?

Answer 4

z2=x2+y2

Lines if k =0

Question 5

What is the equation for a elliptic paraboloid and what are its axis?

Answer 5

z=x2+y2

variable to first power is axis

Question 6

What is the formula for a hyeprboloid of one sheet and what is its axis of symmetry?

Answer 6

x2+y2-z2=1
variable with negative coefficient

Question 7

What is the formula for a hyperbolic paraboloid?

Answer 7

z=x2-y2

Question 8

What is the formula for a hyperboloid of two sheets?

Answer 8

-x2-y2+z2=1