Parametric Surface

Introduction

Parametric surfaces map a pair of parameters $(u, v)$ into 3D coordinates $(x, y, z)$ using mathematical expressions. They produce smooth, mesh-like geometry ideal for visualizing formulas, exploring shapes, and creating polished motion graphics.

Purpose of this guide: Help users understand the UI options, common usage patterns, and performance pitfalls.

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Interface Overview

Controls are organized into four main panels in the Inspector (right sidebar):

1. Geometry: Define the mathematical shape (x, y, z equations) and spatial transform.
2. Parameters: Define custom constants (e.g., a, b) to control the shape dynamically.
3. Settings: Configure the simulation domain, resolution, motion speed, and rendering mode.
4. Appearance: Control visual style (dots, lines, colors, trails).

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Configuration Details

1. Geometry (Shape Definition)

• Equations:
  • x(u,v), y(u,v), z(u,v): Mathematical formulas defining the surface.
  • Supports standard math functions (sin, cos, exp, etc.) and constants (PI, TWO_PI).
  • Example: x = (R + r * cos(v)) * cos(u) (Torus).
• Transform:
  • scale_x/y/z: Multipliers for the final coordinates.
  • offset_x/y/z: Translation of the entire surface.

2. Parameters

• Define custom variables here (e.g., name: a, value: 2.5).
• Use these variables inside your Geometry equations. Changing the value here updates the shape in real-time.

3. Settings (Domain & Simulation)

• Render Mode:
  • CPU: High precision, supports Trails and Line Weight. Best for high-quality video export.
  • GPU: High performance, supports millions of particles. Trails are disabled. Best for real-time preview and massive point clouds.
• Simulation:
  • speed_u, speed_v: How fast the parameters $u$ and $v$ evolve over time (creates flow/movement).
• Domain:
  • u_min, u_max, v_min, v_max: The range of the parameters.
  • u_segments, v_segments: Grid resolution (number of points). Higher values = smoother surface but higher cost.
  • periodic_u, periodic_v: Connects the edges of the domain (e.g., for closed loops/tubes).
  • flowing_u, flowing_v: Flow Mode. Checked = Continuous seamless flow (ideal for perfect loops). Unchecked = Ping-pong mode (particles bounce back at edges).

4. Appearance

• Style:
  • dot_size: Diameter of the particles/vertices.
  • line_weight: Thickness of the wireframe lines (CPU only).
• Trails (CPU Mode Only):
  • show_trails: Enable motion trails.
  • fade_trails: Trails fade out over time.
  • trail_lifetime: Duration of trails in seconds. Warning: Values > 15s can consume excessive memory.
• Palette:
  • Manage colors for the gradient.
  • Use the Randomize button to generate harmonious palettes (Monochromatic, Analogous, Complementary, etc.).

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Performance & Best Practices

Goal	Recommended Settings
High Quality Video	CPU Mode. segments ~50-100. trail_lifetime 5-10s.
Real-time Flow	GPU Mode. segments 200+. speed_u/v > 0.
Smooth Surface	Increase u_segments and v_segments.
Closed Shapes	Enable periodic_u or periodic_v (e.g., for spheres/toruses).

Troubleshooting:

• Laggy Interface: Lower trail_lifetime or switch to GPU Mode.
• No Trails: Check if you are in GPU Mode (trails are disabled). Switch to CPU.
• Broken Mesh: Check periodic settings if the mesh edges look torn.

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