VectorPaint parabola diagrams: compare focus distance and perpendicular directrix distance

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

A U-shaped curve is not automatically the specified parabola. Define a focus and directrix, calculate five samples and draw an approximate curve. Equality concerns distance to a point and perpendicular distance to a line, not equal-radius circles. Assume no automatic locus solving, equation snapping or optical reflection simulation in VectorPaint.

Define focus and directrix first

Set mathematical focus F=(0,1), directrix y=−1 and vertex (0,0). The OpenStax parabola definition uses equal distances to focus and directrix. Here x²=4y, hence y=x²/4. List focus, line and vertex separately; the focus is no circle centre. The definition covers theoretical points, while finite samples cannot prove an arbitrary smooth curve satisfies it. Keep mathematical units separate from later pixels and view zoom.

Calculate five exact samples

Use x=−4,−2,0,2,4 with y=4,1,0,1,4. Symmetry about the vertical axis does not excuse omitting one side’s checks. Record all five coordinate pairs and separate labels from positions. At x=2, y=2 would violate the equation despite a smooth-looking curve. These are original samples, not measurements of antennas or bridges. For an added point x=1 use y=1/4, not a guessed average between neighbouring samples.

Use perpendicular distance to the directrix

At (2,1), distance to F is 2 and perpendicular distance to y=−1 is also 2. At (4,4), focus distance √(16+9)=5 equals line distance 4+1=5. The vertex gives 1 each; negative-x partners match likewise. Write both distances per point rather than diagonally connecting to an arbitrary location on the line. The directrix is infinite: its drawn segment is only a display extent, so visible endpoint caps do not define distance. Preserve signs, radicals and squared terms; resolve uncertain notation before proceeding. Also distinguish focus-to-directrix distance 2 from vertex-to-focus distance 1; do not combine them under one ambiguous focal-distance label.

Map through one canvas scale

Map the mathematical origin to canvas (600,800) at 100 units per mathematical unit: X=600+100x, Y=800−100y. Samples are (200,400),(400,700),(600,800),(800,700),(1000,400); F is (600,700) and the directrix Y=900. The vertex lies below F, with the directrix still lower and the curve opening upward on screen. Vertical-only compression invalidates equal-distance measurement under one unchanged scale. Label approximate placement and distinguish view zoom from object deformation, without an assumed equation-plotting control.

Keep the curve’s approximate status

Join samples in x order, first checking a polyline, then smoothing a copy approximately with a few handles. Avoid a sharp vertex without replacing the whole curve by a semicircle. Correct samples do not make every Bézier point equidistant; retain the approximation caption. An added x=1 sample has y=1/4, not the midpoint (1,1/2) of (0,0)/(2,1). Check added distances independently. Keep the polyline reference. Display limits do not terminate the mathematical domain; extensions indicate continuation only.

Reject a wrong point with both distances

Test (2,2) separately: focus distance √5≈2.236 differs from perpendicular line distance 3. Record both rather than only calling it too high, preserving correct (2,1). Restoring the sample also requires checking stray handles that distort neighbouring curve segments. This rejects one candidate, not every off-curve point. √5 is not 5, and the focus’s vertical difference is 2−1. These are external hand checks, not automatic distance validation or rejected-point behaviour by the app.

Retain readable references in output

Save and inspect PNG five points, F and directrix labels. Any retained distance guides need their own explanation and line style distinct from the curve. Hide temporary numbers without deleting definitions. Replace unreadable radicals/signs with explicit words rather than omitting y=−1. If a small curve looks circular, preserve samples/definition instead of making it rounder. Inspect other formats separately and move labels rather than mathematical points. Manual grids export as objects; inspect native grids directly instead of treating one screenshot as every export test.

Review equation, mapping, distances and approximation

For asymmetry inspect x signs; for downward opening inspect canvas subtraction; for unequal distances inspect y/focus; for a wrong curve through right samples inspect handles and approximation. Recalculate each point after repairs rather than changing labels to fit errors. Retain definition, coordinates, paired distances, mapping, control and approximate status. This explains focus/directrix relationships without all-point numerical proof, automatic analytic geometry or optical simulation, and guarantees no antenna, mirror or structural performance from a static U shape.