Translating Between Cartesian and Polar Basis Vectors
A single point P is described first with x-y coordinates, then with ρ-φ coordinates, linked by a right triangle showing x = ρcosφ and y = ρsinφ as physical projections rather than abstract formulas. The animation builds the unit vectors i-hat, j-hat, ρ-hat, and φ-hat, decomposing ρ-hat and φ-hat into Cartesian components visually. It closes with the transformation matrix, explaining each entry as a projection, and verifies invertibility with a φ = 0 sanity check. Useful for students learning coordinate transformations in precalculus or vector calculus.
16:9 · every frame verified for overlaps, spacing and edges before rendering
The prompt that made it
Create a visual titled: “2(d)(i) — Translating Between x-y and ρ-φ” Follow the MASTER INSTRUCTION. Scene 1 Show one point P. Describe it using x and y. Scene 2 Keep P fixed. Describe it using ρ and φ. Scene 3 Draw the right triangle. Show: x = ρ cosφ y = ρ sinφ Do not merely display these equations. Show the horizontal and vertical projections of the ρ arrow. Scene 4 Show the basis arrows: i-hat j-hat ρ-hat φ-hat Scene 5 Show: ρ-hat = cosφ i-hat + sinφ j-hat by physically splitting the arrow into x/y pieces. Show: φ-hat = −sinφ i-hat + cosφ j-hat by rotating the radial arrow 90°. Scene 6 Introduce the transformation matrix. Explain each matrix entry as one projection. Finally show: S × T = identity as: translate → translate back → original. Include a sanity check at φ = 0.
Make the next one in this series
- Animate how velocity and acceleration vectors transform between polar and Cartesian coordinates
- Show the same basis vector derivation for cylindrical coordinates in 3D
- Visualize the Jacobian determinant of the polar transformation as an area-scaling factor
