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Cycloid — VektorCAD Tutorial

This tutorial shows how to construct a cycloid (locus traced by a point on a circle rolling without slipping along a straight line).
We’ll keep Normal thickness for construction (XY/base line, ordinates, helper circles, dimensions) and switch to Thick for the final cycloid curve. We will also use spline to draw the smooth locus and the carc command (center+radius arc that cuts a selected entity) to get accurate intersection points.

Pitch of one cycloid arch = circumference of the generating circle = P = 2πr.

Tutorial Video

Problem Statement

Given a generating circle of radius r (e.g., r = 20 mm). Construct one arch of a cycloid generated by a point on its circumference as the circle rolls on a straight line without slipping. Show clean construction and the final curve with labels and key dimensions.

Objective

- Use VektorCAD commands: line, circle, point, ucs, carc, spline, text.  
- Thickness: Normal for construction/dimensions; Thick for the cycloid curve.  
- Turn Entity Snap ON throughout (Endpoint, Midpoint, Center, Perpendicular recommended).

Step‑by‑Step

1) Setup

  1. From the Format panel, set line thickness to Normal (options: Thin / Normal / Thick).
  2. Turn Entity Snap ON and Snap ON for accurate horizontals and verticals.

2) Draw the Base (Rolling) Circle

  1. On the Circle menu, click Circle.
  2. Specify the circle center on the grid.
  3. In the command prompt, enter 20 for radius.
  4. Use TEXT to label the center point as C.
  5. Label the bottom point of the circle as P (the generating point).

Circle

3) Draw Line PA and Parallel Lines

  1. Pitch formula: P = 2πr.
  2. Example: if r = 20 mm → P ≈ 125.66 mm, rounded to 126 mm.
  3. Draw a horizontal line starting from point P with a length of 126.
  4. Using the Copy command, copy this line to pass through the circle’s center and top quadrant.

Lines

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 to confirm.
  4. For total number of items, enter 12.
  5. Center point: select C.
  6. Enter 360 for full rotation.
  7. 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

6) Draw Horizontal and Vertical Reference Lines

  1. On the status bar, turn Ortho ON.
  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 these verticals with line CB.

Horizontal & Vertical Lines

7) Use carc to Locate Points Pᵢ

For each division i = 1…12:

  1. From the Arc dropdown, click Cutting Arc.
  2. In the command prompt, set radius = 20 (same as the generating circle).
  3. Center: snap to Ci (the shifted circle center after rolling i steps).
  4. Select the corresponding horizontal line to cut.
  5. The arc will mark point Pi, the cycloid point for step i.

Repeat for i = 1…12 to generate all cycloid points.

Cut Arcs

8) Draw the Cycloid

  1. In the Format panel, change line thickness to Thick.
  2. On the toolbar, click Spline.
  3. Pick points in order: starting at P (P₀), then P1, P2, …, P12, finishing at A (P12 ≈ cusp).
  4. Do not close the spline — the cycloid extends only from P to A.

Cycloid

Result Checklist

  • Base line PA drawn; PA = 2πr divided into N equal steps.
  • Starting circle at C, divided into N equal arc parts; level (horizontal) lines drawn from each division.
  • Verticals through Cᵢ erected at each base division.
  • For each i, carc with center = Cᵢ, radius = 20 used to cut the i‑th horizontal → Pᵢ obtained.
  • Spline through P, P₁…P₁₂, A set to Thick.

Variations (Practice)

  • More arches: Extend the base by another pitch and repeat steps to draw multiple cycloid arches.
  • Curtate/Trochoid: If the tracing point is inside the rim by distance k, use radius = r − k in Step 5 when cutting horizontals (curtate trochoid).
  • Prolate/Trochoid: If the point is outside the rim by distance k, use radius = r + k (prolate trochoid).
  • Epicycloid/Hypocycloid: Roll the generating circle on or inside a fixed circle (replace base line with a circle; the Step‑5 cutting uses centers on that circular path).

Commands Recap

  • line — vertical ordinates and horizontal level lines.
  • circle — generating circle at start; helper circles during construction.
  • arraypolar — divide circle into 12 equal parts.
  • point — mark P, A, Cᵢ, and Pᵢ as needed.
  • divide — divide line into 12 equal parts.
  • carc — center at Dᵢ, radius r to cut level lines and get cycloid points.
  • spline — draw a smooth cycloid through the generated points.
  • text — labels and notes.
  • Format — Normal for construction; Thick for the final cycloid.

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.