The object to be rendered can be defined by:

• Under the option "Implicit equation(s)":
  • One implicit surface cartesian equation, with optional restrictions given by one or more inequalities.
  • Two implicit cartesian equations, with optional restrictions, to define a curve.
  • One or more inequalities, to define the region of the space given by the intersection of their solutions.
• Under the option "Revolution":
  • The revolution of a curve given by the intersection of a plane and a surface, with option restrictions given by one or more inequalities, around a line defined as the intersection of two planes.
  • The revolution of a region in a plane, given by the equation of the plane and one or more inequalities, around a line defined as the intersection of two planes.

Different equations/inequations are separated by line breaks (enter key). You may write simultaneous equations (such as f < g < h or f = g ≥ h, for instance).

Examples:

• x^2+y^2+z^2=1
• x^2+y^2+z^2=1
  x^2+y^2<z^2+0.25
• 1 < x^2+y^2 < 2
  -1 < z < 1

You may use cartesian (x, y, z), cylindric (r, θ, z) or spherical (ρ, θ, φ) coordinates and even mix them freely. We are taking θ ∈ (-π/2, π/2).

3Graf is a rendering experiment for implicitly defined surfaces and regions of spaces. Actually, it is the result of a hobby and my personal curiosity.

The rendering combines classic techniques with the result of some of my studies on realtime raytracing. The techniques are still quite experimental and I believe the performance can be greatly improved.

The code for the rendering layer was written in C/C++, with compilation for WebAssembly done with Emscripten. Except for the OpenGL library, its web version WebGl, and the standard C/C ++ libs (and their web ports made by Emscripten), no other third-party library were used. A libary, also by my own, is used internally for parsing, simplification and derivation of the function that defines the surface. This lib probably does not bring anything special compared to the options available in internet, but it was fun to implement it. It is capable of some algebraic simplifications, derivation and optimization for future use in GPU processing.

As a medium-term target, I plan to release an Android and iPhone version, if possible with augmented reality capabilities.

About the author

My name is Leonardo Silvares and I am currently professor at the Universidade Federal Fluminense, Niterói, Rio de Janeiro, Brazil

I have a past in IT development, acting as team leader and full stack leading programmer for financial market, implementing critical mission server systems and client (mobile and desktop) apps.

I started to study Graphics Computing and I am enjoying it more and more. I have some experiments that I intend to make available soon, specifically related to volume rendering.

You can contact me by email leotadeu@gmail.com.

How to help the project

You can help the project by:

• reporting errors in rendering (even one wrong pixels matters);
• reporting errors in interface;
• giving new ideas of example objects;
• suggesting any kind of improvement;
• creating a more friendly/beautiful design for the web interface;
• asking for new features;
• translating into a new language;

I thank to my friends Baylon de Aguiar and Leonardo Stepanov Carlevaris for help with hosting, domain, testing and motivation.

About this version

This is no more the very first version available, but it was still made available only for test and proof of concept. It should not be taken as a final product. It should not be taken as a product at all. Previous version is no more available.