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Cycloid -- VektorCAD Tutorial

This tutorial shows how to construct a cycloid -- the locus traced by a point on a circle rolling without slipping along a straight line.

Theory

A cycloid is generated when a circle of radius r rolls along a straight line. The pitch (one complete arch) equals the circumference of the generating circle: P = 2 x pi x r.

Conventions

  • Normal thickness for construction (base line, ordinates, helper circles, dimensions)
  • Thick for the final cycloid curve
  • carc command (center+radius arc that cuts a selected entity) for accurate intersection points
  • spline to draw the smooth locus
  • Entity Snap ON throughout (Endpoint, Midpoint, Center, Perpendicular recommended)

Tutorial Video

Problem Statement

Construct one arch of a cycloid generated by a point on the circumference of a circle as it rolls on a straight line without slipping.

Given Value
Generating circle radius (r) 20 mm
Pitch (P = 2 x pi x r) approx. 126 mm
Divisions 12 equal parts
Task Construct one cycloid arch

Objective:

Requirement Details
Commands used Line, Circle, Point, UCS, Carc, Spline, Polar Array, Divide, Text
Construction lines Normal thickness
Final cycloid Thick thickness
Entity Snap ON throughout

Step-by-Step

1) Setup

  1. Thickness -- set to Normal (Thin / Normal / Thick)
  2. Entity Snap ON and Snap ON for accurate horizontals and verticals

2) Draw the Base (Rolling) Circle

  1. Click Circle on the toolbar
  2. Specify the circle center on the grid
  3. At the prompt, enter 20 for the radius
  4. Use Text to label the center as C
  5. Label the bottom point of the circle as P (the generating point)

Circle

3) Draw Line PA and Parallel Lines

  1. Calculate the pitch: P = 2 x pi x r = 2 x 3.14159 x 20 = 125.66, rounded to 126 mm

  2. Draw a horizontal line starting from point P with a length of 126

  3. Using the Copy command, copy this line to pass through the circle's center and top quadrant

Lines

Tip

Use the virtual keyboard calculator to compute 2 x 3.14159 x 20 for an accurate pitch value.

4) Divide the Generating Circle into 12 Equal Parts

  1. Draw a line from C (center) to P (bottom point)
  2. From the Copy dropdown, click Polar Array
  3. Select line CP and press Enter

  4. At the prompts:

    Prompt Value
    Total number of items 12
    Center point Select C
    Full rotation 360

  5. Mark intersection points 1' through 11' around the circle

Divisions

5) Divide Line PA into 12 Equal Parts

  1. In the Format Panel, set point size to 1
  2. From the Point dropdown, click Divide
  3. Select line PA and enter 12
  4. Label the division points as 1 through 11

Line Divisions

Note

The number of divisions on the base line must match the circle divisions (12 in this case) so each rolled position corresponds to a point on the circle.

6) Draw Horizontal and Vertical Reference Lines

  1. Turn Ortho ON from the status bar
  2. Draw horizontal lines through points 11', 10', 9', 8', and 7'
  3. Draw vertical lines through points 1 to 11
  4. Mark the new centers as C1 to C11 at the intersections of verticals with line CB

Horizontal and Vertical Lines

7) Use carc to Locate Points Pi

For each division i = 1 to 12:

  1. From the Arc dropdown, click Cutting Arc

  2. At the prompts:

    Prompt Action
    Specify radius 20 (same as the generating circle)
    Specify center point Snap to Ci (shifted circle center after rolling i steps)
    Specify curve to cut Select the corresponding horizontal line

  3. The arc marks point Pi -- the cycloid point for step i

  4. Press Enter to repeat Cutting Arc
  5. Repeat for i = 1 to 12 to generate all cycloid points

Cut Arcs

Tip

The radius is always the same (r = 20) for every cutting arc. Only the center changes to each successive Ci position.

8) Draw the Cycloid

  1. In the Format Panel, switch to Thick line thickness
  2. Click Spline on the toolbar
  3. Pick points in order: P (start), then P1, P2, ... P12, finishing at A
  4. Press Enter to complete -- do not close the spline

Cycloid

Warning

The cycloid is an open curve from P to A. Do not use Close -- just press Enter to end the spline.

Result Checklist

Item Status
Base line PA drawn; PA = 2 x pi x r divided into 12 equal parts
Generating circle at C divided into 12 equal arc parts
Horizontal lines drawn from each circle division point
Vertical lines through each base division; centers C1 to C11 marked
carc with center = Ci and radius = 20 used to cut each horizontal
Spline through P, P1 ... P12, A set to Thick

Variations (Practice)

Variation What to try
Multiple arches Extend the base by another pitch and repeat steps
Curtate trochoid Tracing point inside the rim by distance k -- use radius r - k for cutting arcs
Prolate trochoid Tracing point outside the rim by distance k -- use radius r + k for cutting arcs
Epicycloid Roll the generating circle on the outside of a fixed circle (replace base line with a circle)
Hypocycloid Roll the generating circle on the inside of a fixed circle

Commands Recap

Command Purpose
line Vertical ordinates and horizontal level lines
circle Generating circle at start
arraypolar Divide circle into 12 equal parts
point Mark P, A, Ci, and Pi as needed
divide Divide base line into 12 equal parts
carc Center at Ci, radius r to cut level lines and get cycloid points
spline Draw smooth cycloid through generated points
text Labels and notes
Format Normal for construction, Thick for final cycloid

Export and share

You've constructed a cycloid using equal divisions, helper horizontals/verticals, carc for precise intersections, and spline for a clean final curve. Export to PDF to verify line weights before sharing.