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Control point

As explained in Math for Web Design

A control point is a point that determines the shape and local behavior of a Bézier curve without necessarily lying on the curve.

A control point shapes a Bézier curve without usually sitting on the curve itself. Think of it as a magnet or handle: move it, and the path bends toward it. As the parameter t moves from 0 to 1, each point’s influence changes predictably, producing a smooth path from the fixed start to the fixed end. section 11.1.2

A quadratic Bézier curve has one control point, so that point affects the whole curve. A cubic curve has two: the first shapes how the curve leaves its start, and the second shapes how it approaches its end. This independent start-and-end control makes cubic curves useful for animation timing. section 11.1.3 section 11.1.4

In a CSS timing function, the endpoints are fixed and the two inner control points control motion:

  • P₁ (x1, y1) controls the beginning: x1 changes when acceleration begins, while y1 changes how strongly it starts. section 11.2.2
  • P₂ (x2, y2) controls the ending: x2 changes when deceleration begins, while y2 changes the final approach. section 11.2.2
  • Keep x1 and x2 between 0 and 1. Letting y-values fall below 0 creates anticipation; values above 1 create overshoot. section 11.2.1

Make a toggle finish gently

Use cubic-bezier(0.2, 0.9, 0.1, 1.0) on a transform transition. Its steep early slope produces a fast start, while its flatter ending lets the toggle settle slowly. Control points change progress over time, not the start or final transform values. section 11.2.1 section 11.1.4

Figure 11.2 The cubic Bézier curve defined by cubic-bezier(0.2, 0.9, 0.1, 1.0)

Inspect the plotted curve for cubic-bezier(0.2, 0.9, 0.1, 1.0) to connect the two inner control points with a quick start and gentle finish. section 11.2.1

How Math for Web Design covers it

How deep it goes
Explains quadratic and cubic roles, coordinate effects, ranges, and anticipation or overshoot, without requiring formula memorization.
Examples
Uses a toggle-switch transform transition plus gentle, aggressive, bounce, symmetrical, and slow-start timing curves.
What it assumes
Starts with plain-language Bézier background and does not require memorizing complex formulas.
What you can do after
Apply it via CSS cubic-bezier() to tune acceleration, deceleration, anticipation, and overshoot.
How it connects
Connects Bézier curves with quadratic curves, cubic Bézier curves, easing functions, animation progress, and CSS transitions.

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This explainer was generated by Manning's knowledge graph from the text of Math for Web Design. The section links point to the parts of the book it draws on.