Game Dev Math: Ultimate guide to polishing your game!

Explore how sine and cosine drive game oscillations by adjusting amplitude, period, frequency, y offset, and x displacement, and reveal their equivalence up to a pi/2 offset for pulse functions.

Discover why game developers favor radians over degrees and how the unit circle defines radians by arc length on the unit circle, with pi equating to 180 degrees.

Explore how sine and cosine relate to the unit circle and right triangles, with x equals cosine theta and y equals sine theta, enabling rotation in game development.

Refresh your understanding of 2D vectors in game dev, covering magnitude, direction, components, and operations like addition, subtraction, negation, scaling, and unit vectors, with points, distances, and perpendicular vectors.

Apply the dot product to measure vector similarity and projection, producing a scalar that reveals direction, front‑behind relations, and the angle between vectors.

Explore how matrices enable moving between coordinate spaces by applying column-major transformations to 2D vectors, using basis vectors and rotation concepts for world and player spaces.

This lecture demonstrates implementing a pulse effect to flash power-ups as their lifetime nears end, using sine-based oscillation, time, and alpha modulation.

Explore rotation via the unit circle, where x is cosine theta and y is sine theta, forming the rotation matrix. Rotate around the origin, then translate around an arbitrary point.

Implement a screen shake that activates when missiles hit asteroids, updating the shake timer and applying sine and cosine offsets. Ensure asteroid constants enable screen shake.

Learn how to bounce a ball off a wall using vector reflection, with velocity vectors, the wall normal, dot products, and a reflect method in a breakout-like setup.

Normalize values to the 0 to 1 range using min-max scaling (subtract min, divide by max minus min). Apply to animations and games, leveraging unit vectors and dot products.

Learn linear interpolation using l(t) = (1-t)A + tB with t in [0,1], moving from A to B, applicable to colors or positions in a linear, even way.

Explore easing functions and how changing the t parameter within 0 to 1 creates ease-in, ease-out, and smoothstep curves for animations, with quadratic and bounce examples and practical implementations.

Learn uv mapping to apply textures to game objects using uv space, normalize to 0–1, and fetch texels from sprite sheets, enabling rotation and scale without extra images.

Apply alpha blending in 2d games using linear interpolation between destination and source colors. Normalize the source alpha to 0–1 and use evaluate one minus source alpha to blend transparency.

Learn how the animated button class uses a pulse to drive alpha-based highlights, text bouncing, and color flashing between teal and orange, driven by a simple update-time animation loop.

Learn how bilinear filtering smooths textures by interpolating along x and y using neighboring texels, based on UV coordinates and sprite size, including border adjustments to sample correctly.

Implement the smooth step function in the easing stance cbp file by translating the easing functions from the lecture, and reference the formula to ensure accuracy.

Implement a gradient function that smoothly transitions between two colors using gradient params, supporting left-to-right, right-to-left, bottom-to-top, and top-to-bottom directions, applied via the overlay color on animated buttons.

Animate each character of the button title using per-letter data, with three methods (set up, update, draw) and a 0.5 second duration; disable bounce and explore left-to-right and reverse directions.

Rays define a direction from an origin extending infinitely, expressed as origin plus direction times t, derived from vector form of interpolation and used for collision detection and ray tracing.

Explore vector representations of circles for collision detection, using the center and a point difference to set a radius, and extend to spheres while preparing for the quadratic equation.

Apply the quadratic equation ax^2+bx+c=0 to find roots with (-b ± sqrt(b^2-4ac))/(2a), as shown by x^2-1=0 yielding -1 and 1, and see how it intersects rays and circles for collisions.

Explore ray-circle intersections for collisions by substituting the ray into circle equation to form a quadratic, solving for t, and interpreting discriminant (two, one, or no intersections).

Explain how to implement laser ray-asteroid intersections using a circle bound, discriminant, and t values, integrate with ship ammo, and prepare for asteroid splitting in the final assignment.

Implement the asteroid split by the ship laser, drawing two halves that fade and move apart from the hit point using world-space split points and UV space orientation.

Celebrate completing the course and see how math powers game development, with encouragement to re-engage and learn vectors, matrices, and quaternions, plus the free 3D Math Primer book.