Cubic Bezier Curve Generator

Visualize CSS transition timing curves, simulate kinetic animations, and export clean animation variables in real-time

Standard Timing Presets

Coordinate Parameters

Control Point 1 (X1)0.25
Control Point 1 (Y1)0.1
Control Point 2 (X2)0.25
Control Point 2 (Y2)1
Simulation Duration1s

Curve Sandbox & Preview

Generated transition rules
transition: all 1s cubic-bezier(0.25, 0.1, 0.25, 1);

The Mathematical Formulations of Bernstein Polynomials and Cubic Bezier Timing

In computer graphics, physics simulation engines, and front-end layout transitions, standardizing movement curves is critical to achieving realistic, responsive UI interactions. Rather than employing strict linear movements that feel inorganic, browsers utilize parametric curves called **Cubic Bezier transitions**. By translating control point coordinates through algebraic interpolation, this **Cubic Bezier Curve Generator** provides real-time mathematical visualizations and transition scripts.

The Mathematical Calculus of Cubic Bezier Curves

A Cubic Bezier curve is a parametric curve defined by four distinct points: $P_0$ (start point), $P_1$ (first control handle), $P_2$ (second control handle), and $P_3$ (end point). In CSS transitions, $P_0$ is hardcoded to $(0,0)$ representing start time/progress, and $P_3$ is hardcoded to $(1,1)$ representing completion. The coordinates of $P_1 (x_1, y_1)$ and $P_2 (x_2, y_2)$ are customized to define the curve's profile.

The mathematical position $B(t)$ of the curve at any parametric interval $t$ (from 0 to 1) is computed using third-degree Bernstein polynomials:

B(t) = (1 - t)³P_0 + 3(1 - t)²tP_1 + 3(1 - t)t²P_2 + t³P_3

Because CSS timing functions map elapsed time ($X$) to animation progress ($Y$), the browser's render engine must solve the horizontal component for $t$ to determine the current vertical progress $Y$.

Handling Values Beyond the [0, 1] Boundaries: Elastic and Bouncy Physics

While the horizontal parameters ($x_1, x_2$) representing timeline progression are physically capped between $0$ and $1$, the vertical coordinates ($y_1, y_2$) representing progress have no such limits.

Setting the vertical coordinates below $0$ or above $1$ (such as `cubic-bezier(0.68, -0.6, 0.32, 1.6)`) generates **elastic kinetic behaviors**. The animation will briefly regress below its starting point before accelerating past its ending point and snapping back—mimicking physical spring dynamics. This effect is useful for designing tactile, modern UI micro-interactions.

Zero-Network Diagnostics for Design and Animation Assets

Front-end designers and UI developers regularly calibrate exact transition profiles for proprietary software products, enterprise apps, or unreleased layouts. Uploading design parameters to remote, server-logged animation converters leaks structural design choices and branding guidelines.

In alignment with our strict **100% Client-Side Privacy Law**, this generator operates entirely within your browser's private memory sandbox. All coordinate rendering, mathematical calculations, and kinetic animations are computed locally inside client-side RAM, keeping your layout parameters and UI choices confidential.

📈 Interface Animation Pro Tip

To make user interactions feel incredibly snappy, use an asymmetric curve where the initial acceleration is fast and the final deceleration is gradual (such as `cubic-bezier(0.16, 1, 0.3, 1)` or Ease-Out Expo). This provides instant visual feedback to user inputs. Save your custom kinetic profiles to the local **History Log** to build and compare custom animation assets safely.