VectorPaint ellipse-focus diagrams: distinguish semiaxes and the two focal distances

English translation of the Chinese original, translated on 2026-11-07. Original publication dates, attribution, code and version notes are retained.

Ellipse foci are not arbitrary decorative points on an axis. Choose semimajor 5, semiminor 3 and each focus 4 from centre, then check labels through simple arithmetic. This is no automatic geometry proof or measurement certification. Retain drawing and calculations separately and identify manual approximations.

Fix the centre and invented-unit scale

Create a 1200 by 800 pixel canvas centred at (600,400), with 60 pixels per unit. Semiaxes 5/3 become 300/180 pixels, giving full width 600 and height 360; do not label half as full. Draw horizontal/vertical references and identify invented units, not centimetres. Long-axis endpoints are x=300/900 and vertical endpoints y=220/580. Check bounds and retain the coordinate list. Separate labels so moving text cannot alter geometry.

Keep the ellipse concentric with perpendicular axes

Use available sizing controls to match the four endpoints, or verify references and label approximation when dragging. Do not tilt the ellipse while retaining horizontal-5/vertical-3 explanations. Inspect an unfilled outline against endpoints without misleading heavy strokes. Centre is not a bounding-box corner or an arbitrary visually balanced point. Save the geometric baseline before computing foci/segments from the same centre; ordinary objects do not understand these relationships automatically.

Distinguish centre-to-focus four from total focal separation

Here 5²−3²=16 and its square root is 4. Each focus is therefore 4 units/240 pixels from centre: (360,400) and (840,400). Their separation is 8, not 4. Name left/right foci with equal markers, inside the ellipse rather than at axis endpoints. Put arithmetic outside and explain its dependence on the chosen semiaxes. Positioned points do not mean the app derived a formula or proved an arbitrary curve elliptical.

Check the distance sum at the right endpoint

At right endpoint (900,400), distances to left/right foci are 9 and 1 units, sum 10=2×5. The drawn segments partly overlap, so list both values rather than counting visible strokes. Measure endpoint-to-focus, not a route via centre plus focal separation. Keep labels clear through short leaders. This simple check is not proof that every point of a hand-drawn outline satisfies the definition or that output has certified precision.

Use the 3–4–5 relation at the top

At top (600,220), each focus differs horizontally by 4 units and vertically by 3, giving length 5 and sum 10. Symmetry follows this axis-aligned construction, not visual equality as measurement. Temporarily colour segments to verify focus attachments, then restore. The top is not the centre and these are not radii. Correct misplaced endpoints rather than changing arithmetic to fit them. Retain both endpoint/top calculations with explicit object correspondence.

Recalculate foci after changing the minor semiaxis

In a copy, change semiminor 3 to 4 while semimajor remains 5. Centre-to-focus becomes 3, not the old 4. Update height, top/bottom, foci, top segments and labels while centre/major axis stay fixed. Focal separation is now 6; each top distance remains 5 and sum 10. Uniform scaling does not isolate this change, and moving foci alone leaves an obsolete ellipse. Name both versions separately and compare at one unit scale.

Keep calculations and approximation scope in output

Retain semiaxis list, ellipse, foci and both segment checks. In export, identify 5/3/4 versus 5/4/3 before coordinates/labels. Keep markers and text readable without obscuring endpoints. Test backgrounds and manage actual backing objects. Name focal teaching diagram, units and revision and exclude test foci. Reopen SVG/PDF in the receiving tool without assuming it recognises mathematical entities. Precise geometric/measurement use needs separate verification; this image explains its construction and arithmetic only.

Separate semiaxes and full lengths when labels conflict

An 8 centre-to-focus label may confuse total separation with one side. A top segment labelled 3 mistakes vertical difference for hypotenuse; return to 3–4–5. After a minor-axis change, verify ellipse/foci revisions before the sum. Repair bounds rather than moving foci to make a misplaced outline look centred. Recheck full width/height, single focal offset and both sums so each number identifies a defined quantity.