Mastering 2D and 3D Rotations: A Mathematical Odyssey

Rotate a point in 2d by theta using rotation matrix, yielding x1 = x0 cos theta - y0 sin theta and y1 = y0 cos theta + x0 sin theta.

Learn to rotate a point in a 2d plane with desmos, using x0,y0 to obtain x1,y1 via x1 equals x0 cos theta minus y0 sin theta.

Explore how to manipulate a single-variable function on the 2D plane using rotation, amplitude, phase, and crossover adjustments, with examples like x^3, x^2, and sine x, via a rotation matrix.

Explore 3d rotations using rotation matrices around the z axis, the x axis, and the y axis, converting degrees to radians, and visualizing how coordinates transform with a graphical calculator.

Rotate a point in 3D space by composing rotations about the x, y, and z axes with matrix multiplication, exploring orientations and x1 and y1 projections on a 2D view.

Explore how to simulate 3d rotation on a 2d screen by rotating a line defined by two points around the z, x, and y axes, using perspective and multiple points.

Explore 3D rotation of a triangle by mapping a 3D plane onto a 2D screen, rotating the triangle around the x, y, and z axes to visualize its motion.

Explore 3D rotation using a simple cube, deriving and applying the rotation matrix as you rotate around the z, x, and y axes from the origin.

Generate and rotate a surface defined by a two-variable function using parametric equations. Build and center a grid with floor and modulus, then rotate this three-dimensional grid.

Rotate a 3d grid and map the z plane onto the x and y outputs, using six rotation equations and inputs g(t), h(t), and f(x,y) for z.

Explore 2D and 3D rotations as they drive a 3D surface generator. Uncover practical techniques for generating and visualizing surfaces in a mathematical odyssey.

Explore how pyramids transform under 2d and 3d rotations, revealing fundamental geometric principles and spatial reasoning in this mathematical odyssey.

Explore rotating a single-variable function in 3d space, like x squared, around the z, y, and x axes, with origin-centered grids.

Explore rotation of a 3d surface for a function of two variables, with labeled x, y, and z axes, using alpha, beta, gamma in degrees, and a joystick for control.

Explore generating 3d parametric functions by defining x, y, and z as outputs of h(t) and g(t) with parameters a and b, including points, lines, parabola, surfaces, sphere, and helix.

Explore a cube through 2d and 3d rotations within a mathematical odyssey, guiding students toward mastery of rotation concepts.

Explore how parametric art uses 2D and 3D rotations to create dynamic visual patterns across digital compositions.

Celebrate completing the course and consider enrolling in the full course for continued learning. Appreciate what you learned as valuable and prepare to join the next course.