VectorPaint quadratic Bézier diagrams: two midpoint levels locate a curve point

English translation of the Chinese original, translated on 2026-11-13. Original publication dates, attribution, code and version notes are retained.

A Bézier control point usually differs from a point on the curve. Build a quadratic construction diagram in VectorPaint using segments, markers and a manually drawn curve. Verify two interpolation levels before presentation, without assuming a dedicated quadratic-curve button or automatic midpoint proof tool.

Separate controls from points on the curve

Use local mathematical P0=(0,0), P1=(4,8), P2=(8,0), x right and y up, at 70 pixels per unit. Separate control polygon, first-level points, second-level points, curve illustration and labels. The Berkeley quadratic Bézier teaching material explains successive interpolation; these controls are original rather than a copied figure. Reverse screen y mapping if necessary so P1 is above the baseline rather than contradicting the labels.

Keep the control polygon distinct from the curve

Draw P0–P1–P2 as two thin grey segments with distinct control markers; the P0–P2 baseline is not a third control edge. P1’s height 8 is a control location, not a required curve point. Before a smooth arc, check point order so P1 and P2 are not exchanged. Without precise placement, retain a coordinate reference and label the image approximate. An attractive dragged curve cannot verify arbitrary Bézier parameters.

Take the two first-level midpoints at half parameter

At t=1/2 take Q0=(2,4) on P0P1 and Q1=(6,4) on P1P2. Verify that each divides its own segment equally, then connect Q0 and Q1 horizontally. This completes only level one; neither is the desired curve point. Taking the midpoint of P0P2 gives (4,0), a baseline point rather than the two-level result. Mark that wrong construction separately and explain its source instead of overwriting the correct points.

Take the second-level midpoint to get four four

The midpoint R of Q0Q1 is (4,4), the curve point at t=1/2. P1=(4,8) shares x but is four units higher; do not move R onto it. Label separate leaders as curve point and control point. This is a parameter midpoint, not a general proof of equal half arc lengths. The symmetry of this example cannot establish the claim for asymmetric curves. State which intermediate values were used rather than applying “midpoint” indiscriminately to different objects.

Calculate quarter and three-quarter points

At t=1/4, interpolation from P0 toward P1 gives (1,2), and from P1 toward P2 gives (5,6). Interpolating one quarter from the former toward the latter gives (2,3). At t=3/4, first-level points (3,6) and (7,2) yield (6,3). Audit those with (4,4) and both endpoints. Equal parameter increments are not equal arc-length increments. Calculate before placement, retaining heights 3 versus 4 rather than flattening them for layout.

Report drawing precision separately from point calculations

Use available pen or curve controls to draw a smooth illustration near the five audit points, not a pointed copy of the control polygon. Without precise control/type entry, label it manually approximate. A smooth line through five points alone does not prove the entire specified quadratic. Keep interpolation lines and values as the checkable construction; the arc aids reading. P1 is not a required on-curve anchor, and handles in another Bézier representation cannot all be identified with these three controls. Parameters zero and one should return P0 and P2, an additional endpoint check. If the drawn arc misses an endpoint, repair it before judging intermediate smoothness; a plausible peak alone cannot validate the whole diagram.

Troubleshoot stopping after one level and screenshot estimates

If notes end at (2,4), check for stopping after level one. For (4,0), check endpoint averaging; for (4,8), check substitution of P1. Retain the source of each error instead of blaming software from a blurred screenshot. After export inspect R and labels: a misplaced (4,4) leader requires label repair, not moving points. Use words as well as colour to distinguish control polygon and curve in grayscale. Never invent unseen software coordinates; preserve exact calculations and approximate placement separately.

Reopen and audit both levels and five points

Save, leave and reopen. Hide the smooth arc and recompute both half-parameter levels from the three controls before auditing quarter/three-quarter points. Restore the illustration and inspect endpoints, peak height and approximation label. Hide unwanted guides in delivery but keep an audit copy. Open output to verify coordinates and retain an editable source. Another reader should reconstruct (2,3), (4,4), (6,3) and explain why P1 is off the curve, rather than treating a smooth-looking arc as exact evaluation of every parameter.