Lecture 21: Polar coordinates and the shadow

x, y, z or ρ, ψ, θ

Pick a coordinate system, type its ranges, then walk a point in three steps. The 3D axes and the region stay on the figure. Earlier construction is kept as a dashed shadow. ψ is the angle from the positive z-axis (the book’s φ). You can type pi or sqrt(2).

Coordinate system

Range for x, y, z

x from

x to

y from

y to

z from

z to

Allowed: pi, e, sqrt, sin, cos. r = √(x² + y²), ρ = √(r² + z²), x = ρ sinψ cosθ.

Step 1: r, θ on the xy-plane

Move r and θ. The point stays on the xy-plane. The 3D region stays above it.

Drag to rotate

region and xy-shadow current r, θ, z ρ earlier step

Move the point

Radius r

Angle θ

Lecture 21: Polar coordinates and the shadow

x, y, z or ρ, ψ, θ

Step 1: r, θ on the xy-plane

Move the point

What this step is doing