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Parametric Rose

3D Mathematical Botanical Surface

Live Drawing Engine

Drawing Progress 0%

Drawing Speed 0.15x

Loop Drawing Tracer Spark

Presets & Variations

Rendering Mode

Color Themes

Surface Geometry

Petal Frequency ($k$) 3.60

Spiral Scale ($s$) 8.00

Petal Cupping ($c_1$) 2.05

Petal Profile ($c_2$) 1.28

Petal Flare Opening 1.00

Spiral Petal Layers ($\theta_{\max}$) 16.0π

Dynamic Flutter & Bloom

Bloom State 1.00

Organic Wind Flutter 0.02

Mesh Grid Density 600×50

LIVE PLOTTING θ = 0.00π | φ = 1.571 rad | (x, y, z) = (0.0, 0.0, 0.0)

$$\varphi(\theta) = \frac{\pi}{2} e^{-\theta / 8\pi}$$

$$X(\theta) = 1 - \frac{1}{2}\left( \frac{5}{4} \left( 1 - \frac{\theta \bmod 2\pi}{\pi} \right)^2 - \frac{1}{4} \right)^2$$

$$(x, y, z) = \left( r \sin\theta,\, r \cos\theta,\, X(x \cos\varphi - y \sin\varphi) \right)$$

Paul Nylander's Rose Surface

Unlike traditional 2D rose rhodonea curves, this 3D parametric surface maps a continuous sheet $(x, \theta)$ into a multi-layered, organic blossoming rose through trigonometric modulation, exponential spiral decay, and polynomial petal cupping.

1. Spiral Elevation Angle $\varphi(\theta)$

$$\varphi(\theta) = \frac{\pi}{2} e^{-\frac{\theta}{8\pi}}$$

Controls the angle of inclination of petals as the spiral winds inward to outward. The exponential decay $e^{-\theta / 8\pi}$ causes inner petals at negative $\theta$ to stand upright into a tight bud, while outer petals curl gently downward.

2. Petal Serration & Modulo $X(\theta)$

$$X(\theta) = 1 - \frac{1}{2}\left( \frac{5}{4} \left( 1 - \frac{(k\theta) \bmod 2\pi}{\pi} \right)^2 - \frac{1}{4} \right)^2$$

Creates the scalloped periodic petal lobes. The quadratic waveform repeats every $2\pi / k$ radians, carving out smooth distinct petal edges along the spiral perimeter.

3. Petal Cupping & Height Function $y(x,\varphi)$

$$y(x, \varphi) = 1.95653\, x^2 (1.27689 x - 1)^2 \sin\varphi$$

A 4th-degree polynomial in radial distance $x \in [0, 1]$. It creates the natural spoon-like bowl curvature and outward lip flaring observed in real rose petals.

4. 3D Spatial Coordinate Mapping

$$\begin{aligned} r &= X(\theta) \cdot (x \sin\varphi + y \cos\varphi) \\ \vec{P}(x, \theta) &= \begin{pmatrix} r \sin\theta \\ r \cos\theta \\ X(\theta) \cdot (x \cos\varphi - y \sin\varphi) \end{pmatrix} \end{aligned}$$

Rotates the 2D cross-section into cylindrical coordinates around the vertical stem axis $z$, creating the dense, interconnected layered blossom.

Interaction Guide

• Rotate: One finger drag / Left mouse click
• Zoom: Two finger pinch / Scroll wheel
• Pan: Two finger drag / Right mouse click
• Live Draw: Tap the Draw button in the bottom dock.