Cycloid Calculator

How Cycloid Calculator Works?

A cycloid is the curve traced by a point on the circumference of a circle as the circle rolls along a straight baseline without slipping. The calculator uses the generating-circle radius R and a rotation parameter θ to determine the selected point on one arch. The angle is interpreted in radians within the mathematical formulas, while the calculator accepts degrees, radians, or turns as supported input formats.

Here, x is the horizontal coordinate measured from the starting cusp and y is the height above the baseline. One complete arch is generated as θ runs from 0 to 2π.

The tangent properties come from differentiating the coordinate equations. The tangent slope is $\frac{dy}{dx}=\frac{\sin (\theta )}{1-\cos (\theta )}=\cot (\frac{\theta }{2})$. The tangent angle relative to the horizontal is $\alpha =\frac{\pi }{2}-\frac{\theta }{2}$ for points within the open interval of an arch. The radius of curvature and curvature are calculated using $\rho =4R\sin (\frac{\theta }{2})$ and $\kappa =\frac{1}{\rho }$, respectively.

Practical Use Cases

Students: Students can use the calculator to verify parametric-curve exercises, compare coordinate values at different rotation parameters, and confirm important cycloid results such as the complete arch width, maximum height, total arc length, and enclosed area.

Mathematicians: Mathematicians can investigate the relationship between the cycloid's parameter, tangent behavior, curvature, and arc length. The outputs make it convenient to examine special points such as the cusp, the midpoint of the arch, and intermediate positions.

Engineers: Engineers can estimate geometric dimensions and local curvature when evaluating cycloidal profiles for motion systems, cam mechanisms, paths, or rolling-contact designs. The radius of curvature and tangent angle are especially useful for checking shape and directional changes.

Designers and CAD users: Designers can use the coordinate, width, height, and area results to establish a cycloidal profile before constructing a drawing or model. The selected radius and rotation parameter provide a straightforward way to explore the geometry of a partial or complete arch.

Accuracy and Limitations

The calculator evaluates the standard no-slip cycloid equations using the entered radius and rotation parameter, so results are mathematically accurate to normal floating-point and displayed-rounding precision. The parameter is restricted to one complete arch, from 0 to 360 degrees or its equivalent representation in the other supported angle formats. Tangent slope and tangent angle are not finite at the cusps where θ equals 0 or 2π; the radius of curvature approaches zero there and curvature becomes unbounded. The formulas also assume an ideal rolling circle, a perfectly straight baseline, and a point located on the circle's circumference, so they do not model slipping, clearance, thickness, manufacturing tolerances, or non-cycloidal profiles.

Quick Examples