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Ellipse (Eccentricity Method) -- VektorCAD Tutorial

This tutorial shows how to construct an ellipse from its focus, directrix, and eccentricity e using VektorCAD.

Theory

An ellipse is the locus of points P such that PF = e x PD, where F is the focus, PD is the perpendicular distance from P to the directrix line, and 0 < e < 1.

Conventions

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

Tutorial Video

Problem Statement

Construct an ellipse when the distance of the focus F from the directrix is equal to 50 mm and eccentricity is e = 2/3.

Given Value
Distance (Focus to Directrix) 50 mm
Eccentricity (e) 2/3
Task Construct ellipse by eccentricity method

Objective:

Requirement Details
Commands used Line, Point, UCS, Carc, Spline, Array, Mirror, Text
Construction lines Normal thickness
Final ellipse 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. Using the mouse cursor, 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

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

Directrix

3) Mark Focus F and Divide CF into 5 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 ( | ) to make division points visually distinct

  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 5 and press Enter

    This divides CF into 5 equal parts (sum of numerator + denominator of eccentricity: 2 + 3 = 5).

    Focus Point

Tip

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

4) Draw VE on the Third Division Point from C

  1. Start Line: Click Line and select the third division point from C as the starting point

  2. Set Distance Option: At the prompt Specify next point [Angle/Distance/Undo], choose Distance

  3. Fix Line Length: Click point F to specify the length equal to VF

  4. Set Angle: At the prompt, type 90 to draw the line perpendicular to CF

  5. Complete Line: Press Enter to finish

  6. Label Points: Use Text to label start as V and end as E

Vertex Line

5) Draw Line CE

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

CE Line

6) Draw Line 1-1'

  1. Turn Ortho ON from the status bar
  2. Draw a line starting between V and F and extend it so it crosses well over line CE
  3. Turn Ortho OFF (no longer needed)
  4. Use Text to label the start as 1 and the intersection with CE as 1'

Line 1-1'

7) Copy Line 1-1' Using ARRAYRECT

  1. From the Copy dropdown, click Array
  2. Select line 1-1' and press Enter

  3. At the prompts:

    Prompt Value
    Number of columns 15
    Number of rows 1
    Base point Select point 1
    Column spacing 10 (mm apart)
    Row spacing 1

Line Array

8) Trim the Extended Lines

  1. Click Trim on the toolbar
  2. At the prompt, select all portions of the vertical lines above line CE

Trim Lines

9) Draw Arcs to Cut the Lines

  1. From the Arc dropdown, click 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

10) 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 mirror prompts:

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

Mirror Lines

11) Draw the Final Spline

  1. In the Format Panel, switch to Thick line thickness
  2. Click Spline on the toolbar
  3. Starting from point V, pick all line/arc intersection points in clockwise order
  4. To close the spline, click Close (do not re-select V)

Ellipse

Tip

Picking points in consistent clockwise or counter-clockwise order ensures a smooth spline without crossings.

12) Draw Tangent and Normal

Step A -- Pick a point on the ellipse:

  1. Click the Point menu, then right-click and select Nearest Context Menu

  2. Hover near the ellipse and pick a point. Mark it as S and draw line FS Nearest Point

Step B -- Construct auxiliary line:

  1. From the Line dropdown, choose Perpendicular
  2. Select line FS
  3. Start at F and extend it to cross the directrix AB Auxiliary Line

Step C -- Draw the tangent:

  1. From the Line menu, pick the intersection point on directrix AB
  2. Choose Angle and select point S for direction
  3. Pick a point beyond S to complete the tangent, then press Enter Tangent

Step D -- Draw the normal:

  1. From the Line dropdown, choose Perpendicular
  2. Select the tangent line
  3. Start at S and pick a point outside the ellipse Normal

Result Checklist

Item Status
Directrix d and Focus F labeled
Axis drawn perpendicular to directrix through F
Offset lines parallel to directrix at distances d
carc used with center = F and radius = e x d to cut offset lines
Closed spline passes smoothly through intersection points
Final ellipse in Thick; construction in Normal
Labels and sample dimensions added

Variations (Practice)

Variation What to try
Different eccentricity Rebuild with e = 1/2 or e = 3/4
Different focus distance Move focus closer/farther from the directrix
Quadrant + Mirror Generate only one quadrant and mirror across principal axes
Compare methods Draw from major/minor axes (conjugate-rectangle method) and compare

Commands Recap

Command Purpose
line Directrix, axis, parallels (offset lines)
point Mark focus F, foot N, and division points
ucs Place/align axes to simplify perpendicular offsets
arrayrect Copy array of lines with equal spacing
trim Trim extra line portions
carc Center + radius arc to cut offset lines and reveal intersection points
mirror Mirror entities from one side to the other
spline Fit smooth closed curve through points (the ellipse)
text Labels and notes
Format Normal for construction, Thick for final curve

Export and share

You've drawn an ellipse by the eccentricity method using carc to locate accurate points and spline to produce a clean curve. Export to PDF to verify and share your work.