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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.
We’ll use Normal thickness for all construction/dimensions and switch to Thick for the final curve. We’ll also use carc (center+radius arc that cuts a selected entity) to get precise intersection points quickly, and spline to draw a smooth parabola.

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 (e = 1).
The vertex V lies halfway between the focus and the directrix along the axis (the perpendicular from F to the directrix). The focal distance FV = p, and the gap between focus and directrix equals 2p.

Tutorial Video

Problem Statement

To construct a parabola, when the distance of the Focus from the directrix is 50 mm.

Task: Construct the parabola by the focus–directrix method and annotate key features (vertex, axis, sample distances).

Objective

- Use VektorCAD commands: line, point, ucs, carc, spline, array, text.  
- Keep Normal thickness for construction lines and dimensions; switch to Thick for the final parabola.  
- Turn Entity Snap ON throughout.

Step‑by‑Step

1) Setup

  1. Thickness → Normal (Thin / Normal / Thick). point
  2. Entity Snap ON point

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: On the toolbar, click Line to start the line command. Using the mouse cursor, draw a vertical line to represent the directrix.

  3. Label the Directrix: On the toolbar, click TEXT and place labels A (at the top end) and B (at the bottom end) of the vertical line.

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

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

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

  7. Disable Snap: Turn Snap OFF once the origin is set, as it is no longer required.

image

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

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

  2. Mark the Focus: On the toolbar, click Point. In the Specify point command prompt, type 50 to place the focus point F at 50 mm from C.

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

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

  5. Divide the Segment CF:

    1. On the Point menu, click Divide.
    2. At the command prompt Select object to divide [ Between ], type or click Between.
    3. Specify first point: Click the start point C on the axis.
    4. Specify second point: Click the focus point F.
    5. Enter the number of segments: Type 2 and press Enter.

This divides line CF into 2 equal parts (because eccentricity = 1).
6. Mark it as V

Vertex

4) Draw Line VE

  1. On the Line menu, click Line and select point V as the starting point.
  2. At the command 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 the command.
  6. Use the TEXT command to label the endpoint as E.

Line VE

5) Draw Line CE

  1. On the Line menu, 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. On the status bar, turn Ortho ON.
  2. Draw vertical lines passing through axis CD and line CE.
  3. Turn Ortho OFF when done.
  4. Use the TEXT command 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:
    1. Specify radius: Select points 1 and 1'.
    2. Specify center point: Select point F.
    3. Specify curve to cut: Select line 1–1'.
  3. An arc will be created, cutting line 1–1'.
  4. Press Enter to repeat the Cutting Arc command.
  5. Repeat the process for the other vertical lines.

Cutting Arcs

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:
    1. Pick mirror reference start: Select point C.
    2. Pick mirror reference end: Select point F.

Mirror

9) Draw the Final Spline

  1. From the Format panel, set line thickness to Thick.
  2. On the toolbar, click Spline.
  3. Starting at point 4', select all line/arc intersection points in counter-clockwise order.
  4. Press Enter to complete the command.

Parabola

Result Checklist

  • Directrix d, Focus F, Axis FN drawn and labeled.
  • Vertex V located at the midpoint of FN.
  • Several offset lines L(d) parallel to the directrix placed at chosen distances.
  • For each L(d), carc with center = F and radius = d used to get intersection point(s).
  • A smooth, open spline passes through the collected points.
  • Final parabola set to Thick; all construction remains Normal.

Variations (Practice)

  • Move the focus closer/farther from the directrix (changes p).
  • Rotate the directrix and re‑align the UCS; repeat the construction at an oblique angle.
  • Generate points only on one side and mirror about the axis.
  • Compare with parabola generated from a rectangle/tangent method (equal angles property).

Commands Recap

  • line — directrix, axis, and parallel offset lines.
  • point — mark F, V, and cut points.
  • ucs — reposition/align axes for easy numeric offsets.
  • carc — center at F, radius = d to cut each offset line and reveal locus points.
  • spline — draw the smooth (open) parabola through the points.
  • mirror — to mirror the lines and arcs in bottom quadrant.
  • text — labels and notes.
  • Format — Normal for construction, Thick for final curve.

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.