# GfsSurface

### From Gerris

A GfsSurface is an oriented surface (in 3D) or an oriented curve (in 2D).

The surface can be defined implicitly using for example:

(x*x + y*y + z*z - 0.1*0.1)

which defines the surface as the set of points of coordinates `(x,y,z)`

such that `x*x + y*y + z*z - 0.1*0.1 = 0`

(i.e. a sphere of radius 0.1 centered on the origin).

The sign of the implicit function defines the surface orientation. The function should be continuous. For example it is not a good idea to do

(x*x + y*y + z*z > 0.1*0.1 ? 1. : -1.)

instead of the definition above.

Surfaces can also be defined explicitly using GTS files. For example:

sphere.gts

When using GTS surfaces in two dimensions, the oriented curve is defined as the intersection of the GTS surface with the `z = 0`

plane.

The surface definition can be followed by an optional parameter block with the following syntax:

{ tx = 0.1 ty = -0.2 tz = 0.4 sx = 2. sy = 1.5 sz = -1 rx = -45 ry = -5.5 rz = 25 scale = 3. flip = 1 twod = 1 }

where `(tx,ty,tz)`

is a translation vector, `(sx,sy,sz)`

is a scaling vector and `(rx,ry,rz)`

is a rotation vector (angles in degrees around each principal direction). Note that the translation is always performed after rotation and scaling.

If the `scale`

parameter is specified, uniform scaling is applied along the three axis.

The `flip`

parameter can be used to flip the surface orientation.

If set to one the `twod`

parameter "flattens" the surface on the `z = 0`

plane (this is used in 3D by the GfsRefineSurface object).

Several simple implicit surfaces are pre-defined:

`ellipse(x,y,a,b)`

- an ellipse (an elliptical prism in 3D) centered on
`(x,y)`

and with semimajor axis`a`

and semiminor axis`b`

.

`sphere(x,y,z,r)`

- a sphere centered on
`(x,y,z)`

and of radius`r`

.

`cube(x,y,z,h)`

- a cube centered on
`(x,y,z)`

and of size`h`

.

## Boolean operations

Boolean operations between implicit surfaces can be used to create more complex objects (a technique also know as Constructive Solid Geometry). Given two implicit surfaces A and B with associated implicit functions `fA`

and `fB`

, the standard boolean set operations can be constructed as:

- Intersection
- A ^ B = {
`MAX (fA(x,y,z), fB(x,y,z))`

} - Union
- A U B = {
`MIN (fA(x,y,z), fB(x,y,z))`

} - Difference
- A - B = {
`MAX (fA(x,y,z), - fB(x,y,z))`

}

For example the following will replicate the example of the wikipedia CSG page

Solid ({ double s = sphere (0, 0, 0, 0.25); double c = cube (0,0,0,0.38); double sIc = MAX (s, c); double cylinder1 = x*x + y*y - 0.12*0.12; double cylinder2 = z*z + y*y - 0.12*0.12; double cylinder3 = x*x + z*z - 0.12*0.12; double cylinderU = MIN (MIN (cylinder1, cylinder2), cylinder3); return MAX (sIc, - cylinderU); })