Math

The small maths toolbox the rest of the engine leans on: vectors, shapes, collision tests and interpolation.

The concept

2D games are mostly vectors and rectangles. Keeping them in one dependency-free module, rather than reaching into SDL types, means gameplay, physics, rendering and the editor all speak the same small set of types.

Vectors

A 2D vector is a pair of numbers that can mean a position, a direction or a displacement. Adding vectors moves things, a scalar multiple stretches them, and its length gives distance or speed. Working with vectors instead of separate x and y variables keeps formulas short and hard to get wrong.

Shapes and overlap tests

Two axis-aligned rectangles overlap exactly when they overlap on both axes, which costs a handful of comparisons. Two circles overlap when the distance between their centres is smaller than the sum of the radii. Cheap tests like these are why games approximate complicated art with simple collision shapes.

Penetration

Knowing that two shapes overlap is not enough to react; you also need to know how to separate them. The minimum translation vector is the shortest move that makes them just touch. For rectangles it is the smaller of the overlaps on the two axes; for circles it lies along the line between the centres.

Interpolation and easing

Interpolating means blending between two values by a fraction t from 0 to 1. Feeding t through an easing function makes motion start slowly, end softly or overshoot, which is what makes UI and animation feel natural. Splines extend this idea to smooth curves through several points. Floating-point rounding means tests on values should allow a small tolerance instead of exact equality.

How ASGE does it

Linear algebra

asge::math::Vec2<T> (and Vec3), built on a shared VecN, with the common aliases Float2, Double2, Int2 and Int642. Arithmetic operators are provided; positions, sizes and mouse deltas throughout the API are Float2 or Int2.

Geometry and collision

Rect and Circle are the two collider shapes. Collision.hpp has the overlap tests (AabbOverlap, CircleOverlap) and PenetrationVector overloads for every rect/circle pairing, returning an std::optional<Float2> that says how to push the first shape out of the second. Physics is built on these.

Interpolation

Easing and Ease(easing, t): Linear, InQuad, OutQuad, InOutQuad, OutCubic; the input is clamped to [0, 1].
Tween<T>: moves a float, Float2 or similar from one value to another over a duration; call Update(dt) each frame and read Value().
CatmullRomSpline: a smooth curve through waypoints, with arc-length sampling so motion along it is at constant speed. PathFollowingSystem and the editor's path preview use it.

math.cpp

using asge::math::Easing;

asge::math::Tween<float> slide( 0.0f, 100.0f, /* seconds */ 2.0f, Easing::OutQuad );

// each frame:
slide.Update( dt );
float x = slide.Value();       // eased value; slide.IsFinished() once the duration has elapsed

Start

Theory

Reference

Examples