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Parabola (Eccentricity Method) -- VektorCAD Tutorial

This tutorial shows how to construct a parabola from a focus and a directrix using the eccentricity definition with e = 1.

Theory

A parabola is the locus of points P such that PF = PD, where F is the focus and PD is the perpendicular distance from P to the directrix line (eccentricity e = 1).

The vertex V lies halfway between the focus and the directrix along the axis. The focal distance FV = p, and the gap between focus and directrix equals 2p.

Conventions

  • Normal thickness for all construction, projection, and dimension lines
  • Thick for the final parabola curve
  • carc command (center+radius arc that cuts a selected entity) to locate precise intersection points
  • spline to draw the smooth parabola

Tutorial Video

Problem Statement

Construct a parabola when the distance of the Focus from the directrix is 50 mm.

Given Value
Distance (Focus to Directrix) 50 mm
Eccentricity (e) 1 (parabola)
Task Construct parabola by focus-directrix method

Objective:

Requirement Details
Commands used Line, Point, UCS, Carc, Spline, Array, Mirror, Text
Construction lines Normal thickness
Final parabola Thick thickness
Entity Snap ON throughout

Step-by-Step

1) Setup

  1. Thickness -- set to Normal (Thin / Normal / Thick) Normal thickness

  2. Entity Snap ON Entity Snap ON

2) Draw the Directrix and Set the UCS Origin

  1. Enable Snap: Turn Snap ON so the mouse pointer aligns precisely with the grid points Snap ON

  2. Draw the Directrix: Click Line on the toolbar. Draw a vertical line to represent the directrix

  3. Label the Directrix: Click Text on the toolbar and place labels A (top) and B (bottom) of the vertical line

  4. Set UCS Origin: Run the UCS command and position the origin directly on the directrix

  5. Label the Origin: Use Text to add label C near the UCS origin

  6. Draw the Axis: With the Line command, draw a line perpendicular to AB to represent the axis. Add label D at the end of the axis

  7. Disable Snap: Turn Snap OFF once the origin is set

Directrix

3) Mark Focus F and Divide CF into 2 Equal Segments

  1. Set Point Style: In the Format Panel, change point size to 2 and select the dot ( . ) style Point Style

  2. Mark the Focus: Click Point on the toolbar. At the prompt, type 50 to place focus F at 50 mm from C

  3. Label the Focus: Use Text to add label F near the focus point

  4. Change Point Style: Set point size to 4 and switch to vertical bar ( | ) for visually distinct division points

  5. Divide Segment CF:

    Step Action
    a Click Divide from the Point menu
    b At the prompt, type or click Between
    c First point: click C on the axis
    d Second point: click focus F
    e Number of segments: type 2 and press Enter

    This divides CF into 2 equal parts (because eccentricity = 1, so the vertex is at the midpoint).

  6. Label the midpoint as V

Vertex

Tip

Use entity snaps like Endpoint and Node for precise selection when clicking points.

4) Draw Line VE

  1. Click Line and select point V as the starting point
  2. At the prompt, choose Distance
  3. Pick point F to define the line length (equal to VF)
  4. Enter 90 for the angle, drawing a vertical line
  5. Press Enter to complete
  6. Use Text to label the endpoint as E

Line VE

5) Draw Line CE

  1. Click Line and draw a line from point C to point E
  2. From the Trim dropdown, choose Extend
  3. Click line CE near point E and pick a point beyond it to extend the line

Line CE

6) Draw Vertical Lines Through Axis CD and Line CE

  1. Turn Ortho ON from the status bar
  2. Draw vertical lines passing through axis CD and line CE
  3. Turn Ortho OFF when done
  4. Use Text to label the intersections as 1-1', 2-2', 3-3', and 4-4'

Vertical Lines

7) Draw Arcs to Cut the Lines

  1. From the Arc dropdown, select Cutting Arc

  2. At the prompts:

    Prompt Action
    Specify radius Select points 1 and 1'
    Specify center point Select point F
    Specify curve to cut Select line 1-1'

  3. An arc is created that cuts line 1-1'

  4. Press Enter to repeat Cutting Arc
  5. Repeat the process for all other vertical lines

Cutting Arcs

Note

For a parabola (e = 1), the arc radius equals the perpendicular distance from the point to the directrix. This is why PF = PD at every intersection.

8) Mirror Lines and Arcs

  1. From the Copy dropdown, click Mirror
  2. Select all vertical lines, arcs, and line CE, then press Enter

  3. At the prompts:

    Prompt Action
    Mirror reference start Click point C
    Mirror reference end Click point F

Mirror

9) Draw the Final Spline

  1. In the Format Panel, switch to Thick line thickness
  2. Click Spline on the toolbar
  3. Starting at point 4', select all line/arc intersection points in counter-clockwise order
  4. Press Enter to complete the command

Parabola

Tip

Pick points in consistent counter-clockwise order for a smooth spline. Unlike the ellipse, the parabola is an open curve -- do not close the spline.

Result Checklist

Item Status
Directrix d, Focus F, and Axis CD drawn and labeled
Vertex V located at midpoint of CF
Offset lines parallel to directrix at chosen distances
carc with center = F and radius = d used for each offset line
Smooth open spline passes through intersection points
Final parabola in Thick; construction in Normal

Variations (Practice)

Variation What to try
Different focal distance Move focus closer/farther from directrix (changes p)
Rotated directrix Rotate directrix and re-align UCS; repeat at an oblique angle
Half construction + mirror Generate points only on one side and mirror about the axis
Compare methods Compare with rectangle/tangent method (equal angles property)

Commands Recap

Command Purpose
line Directrix, axis, and parallel offset lines
point Mark F, V, and division points
ucs Reposition/align axes for easy numeric offsets
carc Center at F, radius = d to cut offset lines and reveal locus points
spline Draw the smooth open parabola through the points
mirror Mirror lines and arcs to the bottom quadrant
text Labels and notes
Format Normal for construction, Thick for final curve

Export and share

You've drawn a parabola by the focus-directrix method using carc to produce accurate equal-distance points and spline for a clean result. Export to PDF and verify line weights before sharing.