Calculate Helix

Online calculator and formulas for calculating a helix curve

Helix Calculator

Calculate Helix Curve

Calculates pitch, curvature, torsion and arc length of a three-dimensional helical curve.

Radius of the cylinder
Height of one turn
Number of turns
Results
Pitch:
Curvature κ:
Torsion ω:
Arc length:

Visualization

Helix curve

The graphic shows a helix as a three-dimensional spiral curve around a cylinder.
The helix is created by simultaneous rotation and translation along the axis.

What is a Helix?

A helix (spiral curve) is a three-dimensional curve that winds spirally around an axis:

  • Rotation + Translation: Simultaneous rotation and displacement
  • Constant pitch: Uniform height gain per revolution
  • Cylindrical form: Winds around an imaginary cylinder
  • Unwrapping: Results in a straight line when unwrapped
  • Applications: Spiral staircase, thread, DNA structure
  • Properties: Pitch, curvature, torsion

Curvature and Pitch

The geometric properties of a helix are described by various parameters:

Pitch k
\[k = \frac{h}{2\pi r}\]

Ratio of turn height to circumference

Curvature κ
\[\kappa = \frac{1}{r(1+k^2)}\]

Measure of the bending of the curve

Torsion and Twist

The torsion describes how strongly the helix twists out of the plane:

Torsion ω
\[\omega = \frac{k}{r(1+k^2)}\]

Measure of spatial twist

Meaning
  • ω = 0: Planar curve
  • ω > 0: Spatial twist
  • Large ω: Strong helical twist

Calculate Arc Length

The arc length is the actual length of the helix curve:

Arc length s
\[s = 2\pi r \sqrt{1+k^2} \cdot t\]

Length for t turns

Derivation
\[\sqrt{(2\pi r)^2 + h^2} \cdot t\]

From circumference and height per turn

Helix Formulas Overview

Parametric Representation of a Helix
\[\begin{align} x(t) &= r \cos(t) \\ y(t) &= r \sin(t) \\ z(t) &= \frac{h \cdot t}{2\pi} \end{align}\]

Parametric equations with parameter t (angle)

Pitch
\[k = \frac{h}{2\pi r}\]

Ratio of turn height to circumference

Curvature
\[\kappa = \frac{1}{r(1+k^2)}\]

Local bending of the curve

Torsion
\[\omega = \frac{k}{r(1+k^2)}\]

Spatial twist

Arc Length
\[s = 2\pi r \sqrt{1+k^2} \cdot t\]

Total length for t turns

Symbols and Notations
  • r: Radius of the cylinder
  • h: Height of one turn
  • t: Number of turns
  • k: Pitch of the helix
  • κ (kappa): Curvature
  • ω (omega): Torsion
  • s: Arc length
  • π: Pi (≈ 3.14159)

Example

Example Calculation
r = 3 h = 0.5 t = 10
1. Calculate pitch
\[k = \frac{0.5}{2\pi \cdot 3} = \frac{0.5}{18.85} \approx 0.027\]

Very gentle pitch

2. Calculate curvature
\[\kappa = \frac{1}{3(1+0.027^2)} \approx 0.333\]

Moderate curvature

3. Calculate torsion
\[\omega = \frac{0.027}{3(1+0.027^2)} \approx 0.009\]

Low twist

4. Arc length
\[s = 2\pi \cdot 3 \sqrt{1+0.027^2} \cdot 10 \approx 188.5\]

Total length of the helix

The Helix in Mathematics and Engineering

A helix (plural: helices) is a three-dimensional curve that winds spirally around an axis. It is created by the combination of uniform rotation around an axis with uniform translation along that axis. The helix is a fundamental geometric object with diverse applications in nature and technology.

Mathematical Definition

A helix can be represented parametrically as:

\[\vec{r}(t) = \begin{pmatrix} r \cos(t) \\ r \sin(t) \\ \frac{ht}{2\pi} \end{pmatrix}\]

Where r is the radius, h is the pitch height per full revolution and t is the parameter (angle).

Geometric Properties

Pitch

The ratio of turn height to circumference determines how steeply the helix rises.

Curvature

Describes how strongly the curve deviates from a straight line.

Torsion

Measures the spatial twist of the curve out of the plane.

Arc Length

The actual length of the wound curve, always greater than the height.

Applications in Practice

Helix structures are found everywhere in nature and technology:

  • Architecture: Spiral staircases, helical ramps, towers
  • Mechanical engineering: Threads, screws, twist drills, springs
  • Biology: DNA double helix, protein structures, snail shells
  • Physics: Magnetic field lines, particle paths in accelerators
  • Chemistry: Molecular structures, crystal lattices
  • Technology: Pipelines, cables, transport screws

Special Properties

Unwrapping

When a helix is "unrolled" on a cylinder surface, a straight line is created.

Constant Pitch

The pitch remains constant over the entire curve, unlike other spirals.

Chirality

Helices can be right-handed or left-handed (right or left thread).

Scalability

Helix structures work at all size scales, from molecular to macroscopic.

Related Curves

The helix belongs to a family of related curves:

  • Archimedean spiral: Spiral in the plane with constant radius growth
  • Logarithmic spiral: Spiral with constant growth factor
  • Conical helix: Helix on a cone instead of cylinder
  • Spherical spiral: Spiral on a sphere surface

Calculation and Optimization

In practice, optimization of helix parameters is often important:

  • Threads: Optimal pitch for strength and functionality
  • Stairs: Comfortable pitch for users
  • Springs: Stiffness and elasticity through helix parameters
  • Drills: Chip removal and cutting performance

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