Ellipse (Eccentricity Method) — VektorCAD Tutorial¶

This tutorial shows how to construct an ellipse from its focus, directrix, and eccentricity e using VektorCAD.
We’ll use Normal thickness for all construction/projection/dimensions and switch to Thick for the final ellipse curve. We’ll also use the carc command (center+radius arc that cuts a selected entity) to locate precise intersection points quickly, and spline to draw the smooth ellipse.

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

Tutorial Video¶

Problem Statement¶

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

Task: Construct the ellipse by the eccentricity method. Show a clean final curve with light construction and add a few reference dimensions/labels.

Objective

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

Step‑by‑Step¶

1) Setup¶

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

2) Draw the Directrix and Set the UCS Origin¶

Enable Snap: Turn Snap ON so the mouse pointer aligns precisely with the grid points.
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.
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.
Set UCS Origin: Run the UCS command and position the origin point directly on the directrix.
Label the Origin: Use the TEXT command to add label C near the UCS origin.
Draw the Axis: With the Line command, draw a new line perpendicular to AB (the directrix) to represent the axis.
Disable Snap: Turn Snap OFF once the origin is set, as it is no longer required.

3) Mark Focus `F` and Divide `CF` into 5 Equal Segments¶

Set Point Style: In the format panel, change the point size to 2 and select the dot ( . ) point style.
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.
Label the Focus: Use the TEXT command to add the label F near the focus point.
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.
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 5 and press Enter.

This divides line CF into 5 equal parts (based on the sum of numerator and denominator of the eccentricity).

Focus Point

💡 Tip: Use entity snaps like End Point and Node for precise selection when clicking points.

4) Draw `VE` on the Third Division Point from `C`¶

Start Line Command: On the Line menu, click Line and select the third division point from C as the starting point.
Set Distance Option: When prompted with
Specify next point [Angle/Distance/Undo],
choose Distance.
Fix Line Length: Click point F to specify the length VF. This sets the line length equal to VF.
Set Angle: At the command prompt Specify the angle, type 90 to draw the line perpendicular to CF.
Complete Line: Press Enter to finish the line command.
Label Points: Use the TEXT command to label the start point as V and the end point as E on the vertical line created.

Vertex Line

5) Draw Line `CE`¶

On the Line menu, click Line and draw a line from point C to point E.
From the Trim dropdown, select Extend.
Click on line CE near point E to extend it, then pick a point beyond E to lengthen the line.

CE Line

6) Draw Line `1–1'`¶

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

Line11

7) Copy Line `1–1'` Using `ARRAYRECT`¶

From the Copy dropdown, click Array.
Select line 1–1' and press Enter.
At the command prompt:
1. Enter number of columns: Type 15 to create 15 copies in the X direction.
2. Enter number of rows: Type 1 (only one row).
3. Specify base point: Select point 1 as the base point.
4. Specify column spacing: Type 10 to space lines 10 mm apart.
5. Specify row spacing: Type 1 (since there’s only one row).

Line Array

8) Trim the Extended Lines¶

On the toolbar, click Trim.
When prompted Specify segments to remove, select all portions of the vertical lines above CE.

Trim Lines

9) Draw Arcs to Cut the Lines¶

From the Arc dropdown, click Cutting Arc.
At the prompt:
1. Specify radius: Select points 1 and 1'.
2. Specify center point: Select point F.
3. Specify curve to cut: Select line 1–1'.
An arc is created that cuts line 1–1'.
Press Enter to repeat the Cutting Arc command.
Repeat the process for all other vertical lines.

Cutting Arcs

10) Mirror Lines and Arcs¶

From the Copy dropdown, click Mirror.
Select all vertical lines, arcs, and line CE, then press Enter.
At the mirror prompts:
1. Pick mirror reference start: Click point C.
2. Pick mirror reference end: Click point F.

Mirror Lines

11) Draw the Final Spline¶

From the Format panel, choose Thick line thickness.
On the toolbar, click Spline.
Starting from point V, pick all line/arc intersection points in clockwise order.
To close the spline, do not select V again — instead, click Close to generate a smooth curve.

Ellipse

12) Draw Tangent and Normal¶

Click the Point menu, then right-click and select Nearest in the context menu.
Hover the mouse near the ellipse and pick a point.
- This point will be the location where the tangent is drawn.
- Mark it as S and draw the line FS.
To construct the auxiliary line:
- From the Line dropdown, choose Perpendicular.
- Select line FS.
- Specify the start point at F and extend it to cross the directrix AB.
Draw the tangent:
- From the Line menu, pick the intersection point on the directrix AB.
- In the command prompt, choose Angle.
- Select point S to specify the tangent direction.
- Pick a point beyond S to complete the tangent, then press Enter.
Draw the normal:
- From the Line dropdown, choose Perpendicular again.
- Select the tangent line.
- Specify S as the start point and pick a point outside the ellipse to finish the normal.

Result Checklist¶

Directrix d and Focus F labeled.
Axis drawn perpendicular to the directrix through F.
Several offset lines parallel to the directrix at distances d.
carc used with center = F and radius = e × d to cut each offset line → intersection points collected.
Closed spline passes smoothly through the points.
Final ellipse Thick; construction Normal.
Optional labels and sample dimensions added.

Variations (Practice)¶

Change the eccentricity to e = 1/2 or e = 3/4 and rebuild the curve.
Move the focus closer/farther from the directrix to see how the shape changes.
Generate only a quadrant of points and mirror across the principal axes to save time.
Compare with an ellipse drawn from major/minor axes (conjugate‑rectangle method) to see how points align.

Commands Recap¶

line — directrix, axis, parallels (offset lines).
point — mark focus F, foot N, and cut points if desired.
ucs — place/align axes to simplify perpendicular offsets.
arrayrect - copy array of lines with equal distance.
trim - trim extra lines.
carc — with center and radius to cut each offset line and reveal intersection points.
mirror - mirror entities from one quadrant to another.
spline — fit a smooth closed curve through the points (the ellipse).
text — labels and notes.
Format — Normal for construction, Thick for final curve.

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