VectorPaint translation and reflection: check each vertex coordinate and orientation

English translation of the Chinese original, translated on 2026-11-09. Original publication dates, attribution, code and version notes are retained.

Copying a shape nearby does not establish a specified translation, and a similar-looking flip does not establish vertex correspondence. Fix triangle coordinates, construct translated/reflected copies and check double reflection. No automatic constraint or proof tool is assumed; verify numbers and labels individually.

Choose an asymmetric triangle and coordinate system

Use A(100,200), B(350,250), C(200,450), closed in that order under rightward-x/downward-y coordinates. Put labels outside, distinct from vertices. The asymmetric triangle makes reflected correspondence visible. Preserve a coordinate-labelled baseline without editing nodes or rounding corners during duplication. If the document uses another unit, establish consistent exercise coordinates rather than equating a screen drag with document-coordinate displacement.

Add the same translation vector to all vertices

Define translation as(+400,+100), giving A′(500,300), B′(750,350), C′(600,550). Check all three old/new differences, not A alone. Move a complete duplicate using available position references; if exact placement is unavailable reconstruct from three points and label precision. Translation neither swaps A/B order nor changes edge lengths. Moving one node while deforming the others is not this translation.

Reflect through x-coordinate and axis distance

Start another copy from the original. Reflect in vertical x=600 using x′=1200−x and unchanged y: A′(1100,200), B′(850,250), C′(1000,450). Originals lie left and results right of the axis, with equal paired horizontal distances. Do not substitute a page edge as mirror centre or reflect the translated copy while retaining original-based results. State that both transformations begin from the same baseline.

Preserve vertex identity instead of renaming by position

After reflection A is right of B; do not exchange labels to preserve a left-to-right reading habit. Following corresponding A,B,C reverses handedness under reflection but preserves it under translation, without relying on colour. Prefix views original/translated/reflected while retaining vertex identities. Text can remain readable rather than being mirrored as mathematical content. After rearranging labels inspect leaders to actual vertices so typography does not corrupt correspondence.

Check three squared edge lengths independently

Original AB displacement(250,50) has squared length65000; AC(100,250) gives72500; BC(−150,200) gives62500. Corresponding edges retain these after translation/reflection. Squared values avoid decimal roots and must not be labelled actual lengths. For mismatch inspect both endpoints and rules instead of scaling the whole shape. Equal edge lengths alone do not establish the specified transformation; correspondence and axis-distance checks remain necessary.

Check that two reflections restore the original points

Reflect again in the same vertical axis, subtracting the reflected x from1200 to recover100,350,200, with unchanged y. Two tool flips do not guarantee this if their reference axes differ. Record both axes; do not use x600 first and a shape-centre axis second. Two identical translations accumulate(+800,+200), not restoration. Compare these checks to distinguish transformation rules rather than button names.

Keep coordinates and shapes consistent in output

Save and inspect a PNG for labels, mirror axis and coordinates. Independently resized views need display scales and original coordinate identities; thumbnail distances cannot compare axis distances. Mark translation arrows as vectors, not rotation arcs, and reflection-pair connectors perpendicular to the axis. Hide unwanted manual grids and verify native guides in actual output. Enlarge margins for small coordinates rather than dropping signs or digits.

Locate faults in baseline, rule and labels

For unequal point differences inspect translation; for changed reflected y inspect axis direction; for wrong identities at correct positions inspect renamed labels. If double reflection fails inspect both axes before blaming controls. Recompute three squared lengths after repairs and retain the initial counterexample. Deliver original/new coordinates, length checks and double-transform outcomes, traceable per vertex, without claiming automatic rigid-motion or arbitrary-transform certification.