# VectorPaint logarithmic spirals: audit radius doubling every quarter-turn
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Canonical: [VectorPaint logarithmic spirals: audit radius doubling every quarter-turn]()
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Published: 2026-11-14T00:00:00+08:00
Modified: 2026-11-14T09:33:58+08:00
Author: Bison
Audit radii one, two, four, eight and sixteen over four quarter-turns, with a separate equal-increment radial control.
English translation of the [Chinese original](), translated on 2026-11-14. Original publication dates, attribution, code and version notes are retained.
Instead of a fixed radial increment as in an Archimedean spiral, multiply radius by the same ratio for equal angular steps. Use five anchors and intermediate samples for an approximate logarithmic-spiral diagram, retaining calculations externally. A few line segments are not an exact curve, and natural resemblance does not establish this model.
## Declare angle units and a multiplicative rule
Declare angle zero and radius 1 on the positive horizontal axis, turning counterclockwise and doubling every 90 degrees. Thus r=2^(angle/90). In radians the denominator becomes π/2; never substitute degree values into the radian form. Use an 1800-square canvas centred at (900,900), scale 40, so radius 16 occupies 640. Separate axes, samples, approximate curve and text, adding a direction arrow. Screen-down y must not reverse the declared growth direction.
## List five radial anchors over one revolution
At 0,90,180,270,360 degrees, radii are 1,2,4,8,16. Mathematical coordinates are (1,0),(0,2),(−4,0),(0,−8),(16,0). Start and end share direction but not position: keep them separate on the positive x axis rather than closing a loop. [Wolfram’s logarithmic-spiral definition]() gives the exponential radial relation. This convenient doubling constant is our example, not source evidence about any shell, growth rate or artwork proportion.
## Add forty-five-degree samples before connecting
Every 45 degrees multiplies radius by √2. Coordinates at 45,135,225,315 degrees are (1,1),(−2,2),(−4,−4),(8,−8). Insert these between axis anchors to reduce the coarse corners of a four-axis polyline. For finer sampling use 22.5-degree steps and radius multiplier the fourth root of 2. Derive every sample from the same formula rather than pulling an outer point inward for smoothness. If fitting with handles, distinguish how closely the displayed curve passes samples from the exactness of the external formula.
## Compare successive ratios and differences
Ratios 2/1,4/2,8/4,16/8 all equal 2, while differences are 1,2,4,8 and are not constant. Record invariant ratio and increasing increments in separate rows. “It widens outward” is insufficient because many curves do that. A half-turn multiplies radius by 4 and a full turn by 16, independently checking the same model with different angle steps. Keep stroke width constant. Visible blank gaps between strokes are not automatically the radial differences or distances measured along arbitrary normals.
## Draw a separate fixed-increment comparison
In a separate area use the same five angles but radii 1,2,3,4,5, adding 1 every quarter-turn. These are equal-angle, equal-increment Archimedean samples, not the logarithmic samples. The third anchor becomes (−3,0) rather than (−4,0), and the endpoint is (5,0). Use identical unit scales and all radius labels. Independently fitting each diagram to its frame would undermine visual comparison. Label geometric versus arithmetic growth with full rules, retaining a substantive mathematical difference from the previous lesson rather than merely changing colours or filenames.
## Check rotation and scaling within the declared rule
For this model, rotate a segment back 90 degrees and divide radii by 2 to recover the preceding segment, because fixed angular increments correspond to fixed ratios. Demonstrate using radius 2 at 90 degrees versus radius 1 at zero, and radius 4 at 180 versus radius 2 at 90. This self-similarity follows the exponential rule, not arbitrary spirals, and does not imply equal curvature along every piece. Scaling by 2 is not an added revolution. Only one revolution is drawn; material beyond 360 degrees is not an existing result.
## Troubleshoot closed ends and reversed screen coordinates
End at (16,0), without returning to (1,0). If filling adds a closing edge, remove fill and inspect the editing state, preserving an open curve. Mathematical 90 degrees lies above the centre at screen (900,820), not below while still labelled counterclockwise. Zoom to ensure the radius-one start is not hidden by the centre marker. For small output simplify labels or enlarge the canvas instead of moving samples as though recalculated. Natural references remain separate observations; this exercise does not establish biological conformity to the chosen ratio or angle.
## Reopen and audit five radii and four ratios
Save, leave and reopen. Without answers recover five radii from axes and arrows, checking 1,2,4,8,16 and four ratios of 2, then sample (1,1) at 45 degrees and (8,−8) at 315. The comparison must retain 1,2,3,4,5 at the same scale. Inspect open ends, direction, minus signs and approximation notes in delivery, leaving untested formats unverified. Readers should calculate radius from angle and distinguish geometric from arithmetic growth, without inferring automatic recognition of natural spirals or infinite-precision function plotting.
## Related tutorials and sources
[VectorPaint Archimedean spirals: increase angle and radius under one rule](), [VectorPaint ellipse-focus diagrams: distinguish semiaxes and the two focal distances]().
Checked on November 14, 2026: VectorPaint public version 1.0.8, requiring iOS / iPadOS 17.0 or later. Scope follows the [official App Store]() and [Apple public lookup](). Manual exercises use controls actually available in your installed version; external calculations, written records and diagrams imply no built-in automatic analysis. No step-by-step physical-device, private-file or all-export-environment testing is claimed.