# VectorPaint reciprocal branches: check xy equals twelve without filling the zero gap

Language: en
Canonical: [VectorPaint reciprocal branches: check xy equals twelve without filling the zero gap](<https://allluckly.cn/en/guides/vectorpaint-rectangular-hyperbola-xy-twelve-two-branches/>)
中文：[中文 HTML](<https://allluckly.cn/guides/vectorpaint-rectangular-hyperbola-xy-twelve-two-branches/>)
English: [English HTML](<https://allluckly.cn/en/guides/vectorpaint-rectangular-hyperbola-xy-twelve-two-branches/>)
Published: 2026-11-15T00:00:00+08:00
Modified: 2026-11-15T09:50:54+08:00
Author: Bison

Draw positive and negative coordinate groups as separate branches, using axes as asymptotes without joining them.

English translation of the [Chinese original](<https://allluckly.cn/guides/vectorpaint-rectangular-hyperbola-xy-twelve-two-branches/>), translated on 2026-11-15. Original publication dates, attribution, code and version notes are retained.

Connecting a reciprocal point list in one stroke can invent a line through the origin. Manually organise xy=12 in VectorPaint, checking products before visible ranges, open paths and labels. Distinguish asymptotes, clipping and true endpoints; a finite drawn segment is not the entire function.

## Fix axes and the viewing window

Use an approximately 1700×1700 canvas, origin at screen (800,800), scale 80 px/unit on both axes and window -8..8 in each direction. Screen y is 800 minus 80 times mathematical y, not plus. Separate axes, samples, paths and notes, placing ticks first. Units are invented. The window limits viewing, not the domain; describe continuation beyond clipping.

## Check each positive sample product

List positive pairs (2,6),(3,4),(4,3),(6,2), each product 12. Add window-edge pairs (1.5,8),(8,1.5). Place them in quadrant I, keeping pairs distinct. Sample (2,6) maps to screen (960,320). If points become collinear, recheck rather than changing them to fit a line. These six samples constrain an approximate drawing but do not prove every intermediate point satisfies the equation.

## Keep both coordinates negative on the other branch

Negate both coordinates to obtain (-2,-6),(-3,-4),(-4,-3),(-6,-2), plus boundary pairs (-1.5,-8),(-8,-1.5). Products remain +12 in quadrant III. (-2,-6) maps to (640,1280). This is half-turn symmetry about the origin, not reflection in x; changing one sign gives -12. Negative coordinates are not negative physical lengths or material areas.

## Use one open path per branch

Order each branch independently by increasing x and use separate open, unfilled paths through samples. Never bridge positive and negative groups or close/fill the gap. Check strokes before smoothing node tangents; control handles are not sample points. A labelled polyline approximation is valid without an invented function plotter. External values and manual placement define this exercise; smooth appearance proves no exact equation throughout.

## Keep zero outside the sample set

From y=12/x, x≠0 and finite nonzero x also gives y≠0. Neither axis belongs to the curve. Approaching an axis is not joining it, and the window gives no origin point. Optional continuation arrows follow each branch end, not a bridge toward zero. Positive product excludes quadrants II/IV. This task fixes +12; a negative constant requires a different arrangement.

## Label values and approximation limits

Label selected sample coordinates without crowding every point with arithmetic. Include xy=12 or y=12/x and x≠0. Mark the viewing window, not solid endpoints implying domain termination. Distinguish axes and paths by stroke/legend rather than merely faint colour. Keep a small product ledger checking both groups. Colours aid grouping but prove no sign condition. Retain sample coordinates separately from control handles for later corrections.

## Troubleshoot signs, screen direction and bridging

For quadrant-IV placement check a single sign flip; for vertical inversion check screen-y subtraction. Split accidentally joined/closed paths rather than hiding an origin bridge with white. Reposition incorrect negative samples instead of reflecting across the wrong axis. Spot-check intermediate products and revise an inaccurate approximation, disclosing finite precision rather than deleting labels. After each change recheck samples and open-path state, retaining the native original.

## Reopen and inspect both branches and the axis gap

Save native work, export PNG and reopen. Inspect separated branches and axes small, then signed samples, products and x≠0 enlarged. Repair an accidental filled region in path settings, not the function definition. Reopen independently selectable branches and editable samples/notes. Success explains product 12, quadrants and zero exclusion, distinguishing clipping from endpoints. It claims neither exact fitting, automatic asymptotes nor universal export validation; recalculate changed constants/scales.

## Related tutorials and sources

[VectorPaint manual line charts: plot coordinates before connecting one dataset](<https://allluckly.cn/en/guides/vectorpaint-coordinate-line-chart/>); [VectorPaint parabola diagrams: compare focus distance and perpendicular directrix distance](<https://allluckly.cn/en/guides/vectorpaint-parabola-focus-directrix-distance/>).

Sources checked on 15 November 2026: the [official VectorPaint App Store listing](<https://apps.apple.com/cn/app/id1587272880>) and [Apple public lookup](<https://itunes.apple.com/lookup?id=1587272880&country=cn&entity=software&timestamp=20261115>) show version 1.0.8, requiring iOS / iPadOS 17.0 or later. Draw manually with published shapes, paths, nodes, layers and text. Conditions, scales and layouts are original exercise arrangements; check installed controls. Keep editable native work and PNG previews. SVG/PDF may retain any bitmap content used. External calculations and static diagrams are not new automatic proof, statistics or solver features. No completed device, physical-object or universal export testing is claimed.

Definition: [OpenStax’s conic-sections chapter](<https://openstax.org/books/calculus-volume-2/pages/7-5-conic-sections>) discusses an xy-type rectangular hyperbola. Twelve and sample coordinates are original; no textbook figure is copied or arbitrary Bézier segment claimed to equal a hyperbola exactly.