olb::sdf Namespace Reference

Functions
template<typename T >
T	mix (T a, T b, T h) any_platform
template<typename T0 , typename T1 , typename T2 >
decltype(T0 {} *T1 {} *T2 {})	clamp (T0 x, T1 a, T2 b) any_platform
template<typename T >
Vector< bool, 3 >	isSymmetric (std::function< T(const Vector< T, 3 > &)> sdf, Vector< T, 3 > center) any_platform
	A rough test for symmetry.
template<typename T , unsigned D>
T	sphere (Vector< T, D > p, T r) any_platform
	Exact signed distance to the surface of circle (in 2D) or sphere (in 3D).
template<typename T >
T	box (Vector< T, 2 > p, Vector< T, 2 > b) any_platform
	Exact signed distance to the surface of two-dimensional cuboid.
template<typename T >
T	box (Vector< T, 3 > p, Vector< T, 3 > b) any_platform
	Exact signed distance to the surface of three-dimensional cuboid.
template<typename T >
T	triangle (Vector< T, 2 > p, Vector< T, 2 > a, Vector< T, 2 > b, Vector< T, 2 > c) any_platform
	Exact signed distance to the surface of two-dimensional triangle.
template<typename T >
T	cylinder (Vector< T, 3 > p, Vector< T, 3 > a, Vector< T, 3 > ba, T baba, T r) any_platform
	Exact signed distance to the surface of three-dimensional cylinder.
template<typename T >
T	cylinder (Vector< T, 3 > p, Vector< T, 3 > a, Vector< T, 3 > b, T r) any_platform
	Calculate signed distance to the surface of three-dimensional cylinder defined by the centers of the two extremities and the radius.
template<typename T >
T	cone (Vector< T, 3 > p, Vector< T, 3 > a, Vector< T, 3 > ba, T baba, T ra, T rb) any_platform
	Exact signed distance to the surface of three-dimensional (capped) cone.
template<typename T >
T	cone (Vector< T, 3 > p, Vector< T, 3 > a, Vector< T, 3 > b, T ra, T rb) any_platform
	Calculate signed distance to the surface of three-dimensional capped cone defined by the centers of the two extremities and the corresponding radius.
template<typename T >
T	ellipsoid (Vector< T, 3 > p, Vector< T, 3 > r) any_platform
	Calculate the NOT EXACT (!) signed distance to the surface of three-dimensional ellipsoid.
template<typename T >
T	torus (Vector< T, 3 > p, Vector< T, 2 > t) any_platform
	Exact signed distance to the surface of a torus placed in the XZ-plane of the coordinate system.
template<typename T >
T	solidAngle (Vector< T, 3 > p, Vector< T, 2 > c, T r) any_platform
	Exact signed distance to the surface of a solid angle which represents a combination of a cone (axis y=0 to y=r) and a sphere.
template<typename T , unsigned D>
Vector< T, D >	translate (Vector< T, D > p, Vector< T, D > origin) any_platform
	Translation: The output of this function is used for the calculation of the signed distance to translated/shifted objects.
template<typename T >
Vector< T, 3 >	flip (Vector< T, 3 > p) any_platform
template<typename T >
T	subtraction (T a, T b) any_platform
	Volume of a is subtracted from b.
template<typename T >
T	unify (T a, T b) any_platform
	Volume of a and volume of b are combined as a new object.
template<typename T >
T	intersection (T a, T b) any_platform
	Volume which is shared by a and b creates a new object.
template<typename T >
T	smooth_union (T a, T b, T k) any_platform
template<typename T >
T	smooth_subtraction (T a, T b, T k) any_platform
template<typename T >
T	smooth_intersection (T a, T b, T k) any_platform
template<typename T >
T	rounding (T a, T r) any_platform
	Computes a layer of a constant thickness around the surface.
template<typename T , bool symmetryCheck = true>
T	elongation (std::function< T(const Vector< T, 3 > &)> sdf, const Vector< T, 3 > &p, const Vector< T, 3 > &h, const Vector< T, 3 > &center=(T(0))) any_platform
	Elongation splits the object in 2 (4 or 8) parts, moves them apart and connects them again The object has to be placed in the origin of the coodinate system.
template<typename T , unsigned D>
T	scale (std::function< T(const Vector< T, D > &)> sdf, const Vector< T, D > &p, T s, const Vector< T, D > &center=(T(0))) any_platform
	Function to scale a geometry The object has to be placed in the origin of the coodinate system.
template<typename T >
T	signedDistanceToPorosity (T signedDist, T eps) any_platform
	Converts signed distance to values for the smooth epsilon boundary.
template<typename T >
bool	evalSolidVolumeFraction (T output[], T signedDist, T eps) any_platform

Function Documentation

box() [1/2]

template<typename T >

T olb::sdf::box	(	Vector< T, 2 >	p,
		Vector< T, 2 >	b )

Exact signed distance to the surface of two-dimensional cuboid.

Parameters

p	point for which the distance to the surface is calculated
b	vector containing half side lengths to describe the cuboid

Definition at line 115 of file sdf.h.

{
  Vector<T, 2> q = abs(p) - b;
  return norm(maxv(q, Vector<T, 2>(0.0))) +
         util::min(util::max({q[0], q[1]}), T());
}

References olb::abs(), olb::util::max(), olb::maxv(), olb::util::min(), and olb::norm().

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box() [2/2]

template<typename T >

T olb::sdf::box	(	Vector< T, 3 >	p,
		Vector< T, 3 >	b )

Exact signed distance to the surface of three-dimensional cuboid.

Parameters

p	point for which the distance to the surface is calculated
b	vector containing half side lengths to describe the cuboid

Definition at line 128 of file sdf.h.

{
  Vector<T, 3> q = abs(p) - b;
  return norm(maxv(q, Vector<T, 3>(0.0))) +
         util::min(util::max({q[0], q[1], q[2]}), T());
}

References olb::abs(), olb::util::max(), olb::maxv(), olb::util::min(), and olb::norm().

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clamp()

template<typename T0 , typename T1 , typename T2 >

decltype(T0 {} *T1 {} *T2 {}) olb::sdf::clamp	(	T0	x,
		T1	a,
		T2	b )

Definition at line 50 of file sdf.h.

            {} * T1 {} * T2 {}) clamp(T0 x, T1 a, T2 b) any_platform
{
  if (x < a) {
    return a;
  }
  else if (x > b) {
    return b;
  }
  else {
    return x;
  }
}

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cone() [1/2]

template<typename T >

T olb::sdf::cone	(	Vector< T, 3 >	p,
		Vector< T, 3 >	a,
		Vector< T, 3 >	b,
		T	ra,
		T	rb )

Calculate signed distance to the surface of three-dimensional capped cone defined by the centers of the two extremities and the corresponding radius.

For the calculation of a cone, the second radius is set to zero

Parameters

p	point for which the distance to the surface is calculated
a	vector containing center of the first circular end of the capped cone
b	vector containing center of the second circular end of the capped cone or the apex of the cone
ra	radius of the first circular end of the capped cone
rb	radius of the second circular end of the capped cone

Definition at line 241 of file sdf.h.

{
  Vector<T, 3> ba   = b - a;
  T            baba = ba * ba;
  return sdf::cone(p, a, ba, baba, ra, rb);
}

References cone().

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cone() [2/2]

template<typename T >

T olb::sdf::cone	(	Vector< T, 3 >	p,
		Vector< T, 3 >	a,
		Vector< T, 3 >	ba,
		T	baba,
		T	ra,
		T	rb )

Exact signed distance to the surface of three-dimensional (capped) cone.

Parameters

p	point for which the distance to the surface is calculated
a	vector containing center of the first circular end of the (capped) cone
ba	vector representing the axis of the (capped) cone
baba	length of the cone height squared

Definition at line 210 of file sdf.h.

{
  T rba  = rb - ra;
  T papa = (p - a) * (p - a);
  T paba = (p - a) * (ba) / baba;
  // x distance to the axis of the cone
  // note: abs() prevents from negative values, e.g. caused by rounding errors
  T x = util::sqrt(util::abs(papa - paba * paba * baba));
  // cax and cay for the distances to the caps
  T cax = util::max(0.0, x - ((paba < 0.5) ? ra : rb));
  T cay = util::abs(paba - 0.5) - 0.5;
  // cbx and cby for the distance to the side wall
  T k   = rba * rba + baba;
  T f   = clamp((rba * (x - ra) + paba * baba) / k, 0.0, 1.0);
  T cbx = x - ra - f * rba;
  T cby = paba - f;
  T s   = (cbx < 0.0 && cay < 0.0) ? -1.0 : 1.0;
  return s * util::sqrt(util::min(cax * cax + cay * cay * baba,
                                  cbx * cbx + cby * cby * baba));
}

References olb::util::abs(), clamp(), olb::util::max(), olb::util::min(), and olb::util::sqrt().

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cylinder() [1/2]

template<typename T >

T olb::sdf::cylinder	(	Vector< T, 3 >	p,
		Vector< T, 3 >	a,
		Vector< T, 3 >	b,
		T	r )

Calculate signed distance to the surface of three-dimensional cylinder defined by the centers of the two extremities and the radius.

Parameters

p	point for which the distance to the surface is calculated
a	vector containing center of the first circular end of the cylinder
b	vector containing center of the second circular end of the cylinder
r	radius of the cylinder

Definition at line 195 of file sdf.h.

{
  Vector<T, 3> ba   = b - a;
  T            baba = ba * ba;
  return sdf::cylinder(p, a, ba, baba, r);
}

References cylinder().

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cylinder() [2/2]

template<typename T >

T olb::sdf::cylinder	(	Vector< T, 3 >	p,
		Vector< T, 3 >	a,
		Vector< T, 3 >	ba,
		T	baba,
		T	r )

Exact signed distance to the surface of three-dimensional cylinder.

Parameters

p	point for which the distance to the surface is calculated
a	vector containing center of the first circular end of the cylinder
ba	vector representing the axis of the cylinder
baba	length of the cylinder height squared

Definition at line 172 of file sdf.h.

{
  const Vector<T, 3> pa   = p - a;
  const T            paba = pa * ba;
  const T            x    = norm(pa * baba - ba * paba) - r * baba;
  const T            y    = util::abs(paba - baba * T {0.5}) - baba * T {0.5};
  const T            x2   = x * x;
  const T            y2   = y * y * baba;
  const T            d    = (util::max(x, y) < T {0})
                                ? -util::min(x2, y2)
                                : (((x > T {0}) ? x2 : T {0}) + ((y > T {0}) ? y2 : T {0}));
  return util::sign(d) * util::sqrt(util::abs(d)) / baba;
}

References olb::util::abs(), olb::util::max(), olb::util::min(), olb::norm(), olb::util::sign(), and olb::util::sqrt().

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ellipsoid()

template<typename T >

T olb::sdf::ellipsoid	(	Vector< T, 3 >	p,
		Vector< T, 3 >	r )

Calculate the NOT EXACT (!) signed distance to the surface of three-dimensional ellipsoid.

Parameters

p	point for which the distance to the surface is calculated
r	vector containing half the length of the principal axes

Definition at line 253 of file sdf.h.

{
  const Vector<T, 3> a(p[0] / r[0], p[1] / r[1], p[2] / r[2]);
  T                  k0 = norm(a);
  const Vector<T, 3> r2(r[0] * r[0], r[1] * r[1], r[2] * r[2]);
  const Vector<T, 3> b(p[0] / r2[0], p[1] / r2[1], p[2] / r2[2]);
  T                  k1 = norm(b);
  return (k0 < 1.0) ? (k0 - 1.0) * util::min(util::min(r[0], r[1]), r[2])
                    : k0 * (k0 - 1.0) / k1;
}

References olb::util::min(), and olb::norm().

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elongation()

template<typename T , bool symmetryCheck = true>

T olb::sdf::elongation	(	std::function< T(const Vector< T, 3 > &)>	sdf,
		const Vector< T, 3 > &	p,
		const Vector< T, 3 > &	h,
		const Vector< T, 3 > &	center = (T(0)) )

Elongation splits the object in 2 (4 or 8) parts, moves them apart and connects them again The object has to be placed in the origin of the coodinate system.

Symmetry is required so that the splitted parts are symmetric (check can be disabled)

Parameters

sdf	signed Distance Function of the object which is altered
p	point for which the distance to the surface is calculated
h	vector containing the information how far the splitted parts are moved apart
center	vector containing the center, to shift the object to the origin of the coordinate system for the calculation of the elongation

Definition at line 386 of file sdf.h.

{
  Vector<T, 3> q       = abs(p - center) - h;
  Vector<T, 3> p_elong = p - center;
  // elongation requires symmetry e.g. for elongation in x --> plane of symmetry = YZ-plane
  Vector<bool, 3> sdfIsSymmetric;
  if constexpr (symmetryCheck) {
    sdfIsSymmetric = isSymmetric(sdf, center);
  }
 
  if (h[0] != 0) {
    p_elong[0] = util::max(q[0], 0.);
    if constexpr (symmetryCheck) {
      if (!sdfIsSymmetric[0]) {
        std::cout << "Warning: symmetry in x is not met" << std::endl;
      }
    }
  }
 
  if (h[1] != 0) {
    p_elong[1] = util::max(q[1], 0.);
    if constexpr (symmetryCheck) {
      if (!sdfIsSymmetric[1]) {
        std::cout << "Warning: symmetry in y is not met" << std::endl;
      }
    }
  }
 
  if (h[2] != 0) {
    p_elong[2] = util::max(q[2], 0.);
    if constexpr (symmetryCheck) {
      if (!sdfIsSymmetric[2]) {
        std::cout << "Warning: symmetry in z is not met" << std::endl;
      }
    }
  }
 
  // Second term needed in case of 3D-Elongation if all parts of Vector q are negative
  return sdf(p_elong + center) + util::min(util::max({q[0], q[1], q[2]}), 0.);
}

References olb::abs(), isSymmetric(), olb::util::max(), and olb::util::min().

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evalSolidVolumeFraction()

template<typename T >

bool olb::sdf::evalSolidVolumeFraction	(	T	output[],
		T	signedDist,
		T	eps )

Definition at line 453 of file sdf.h.

{
  T const halfEps = .5 * eps;
  if (signedDist <= -halfEps) {
    output[0] = 1.;
    return true;
  }
  else if (signedDist < halfEps) {
    output[0] = signedDistanceToPorosity(signedDist, eps);
    return true;
  }
  output[0] = 0.;
  return false;
}

References signedDistanceToPorosity().

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flip()

template<typename T >

Vector< T, 3 > olb::sdf::flip

(

Vector< T, 3 >

p

)

Definition at line 305 of file sdf.h.

{
  return {p[1], p[0], p[2]};
}

intersection()

template<typename T >

T olb::sdf::intersection	(	T	a,
		T	b )

Volume which is shared by a and b creates a new object.

The returned distance is not exact if the signed Distance of both objects points to the part of the surface which is removed for creating the combined object

Parameters

a	signed Distance of the first object
b	signed Distance of the second object

Definition at line 341 of file sdf.h.

{
  return util::max(a, b);
}

References olb::util::max().

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isSymmetric()

template<typename T >

Vector< bool, 3 > olb::sdf::isSymmetric	(	std::function< T(const Vector< T, 3 > &)>	sdf,
		Vector< T, 3 >	center )

A rough test for symmetry.

Definition at line 65 of file sdf.h.

{
  Vector<bool, 3> isSymmetric;
 
  isSymmetric[0] = sdf(Vector<T, 3> {1., 1., 1.} + center) ==
                       sdf(Vector<T, 3> {-1., 1., 1.} + center) &&
                   sdf(Vector<T, 3> {1., -1., 1.} + center) ==
                       sdf(Vector<T, 3> {-1., -1., 1.} + center) &&
                   sdf(Vector<T, 3> {1., 1., -1.} + center) ==
                       sdf(Vector<T, 3> {-1., 1., -1.} + center) &&
                   sdf(Vector<T, 3> {1., -1., -1.} + center) ==
                       sdf(Vector<T, 3> {-1., -1., -1.} + center);
  isSymmetric[1] = sdf(Vector<T, 3> {1., 1., 1.} + center) ==
                       sdf(Vector<T, 3> {1., -1., 1.} + center) &&
                   sdf(Vector<T, 3> {-1., 1., 1.} + center) ==
                       sdf(Vector<T, 3> {-1., -1., 1.} + center) &&
                   sdf(Vector<T, 3> {1., 1., -1.} + center) ==
                       sdf(Vector<T, 3> {1., -1., -1.} + center) &&
                   sdf(Vector<T, 3> {-1., 1., -1.} + center) ==
                       sdf(Vector<T, 3> {-1., -1., -1.} + center);
  isSymmetric[2] = sdf(Vector<T, 3> {1., 1., 1.} + center) ==
                       sdf(Vector<T, 3> {1., 1., -1.} + center) &&
                   sdf(Vector<T, 3> {-1., 1., 1.} + center) ==
                       sdf(Vector<T, 3> {-1., 1., -1.} + center) &&
                   sdf(Vector<T, 3> {1., -1., 1.} + center) ==
                       sdf(Vector<T, 3> {1., -1., -1.} + center) &&
                   sdf(Vector<T, 3> {-1., -1., 1.} + center) ==
                       sdf(Vector<T, 3> {-1., -1., -1.} + center);
 
  return isSymmetric;
}

References isSymmetric().

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mix()

template<typename T >

T olb::sdf::mix	(	T	a,
		T	b,
		T	h )

Definition at line 44 of file sdf.h.

{
  return b * (1.0 - h) + a * h;
}

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rounding()

template<typename T >

T olb::sdf::rounding	(	T	a,
		T	r )

Computes a layer of a constant thickness around the surface.

Parameters

a	signed Distance of the object
r	layer thickness

Definition at line 372 of file sdf.h.

{
  return a - r;
}

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scale()

template<typename T , unsigned D>

T olb::sdf::scale	(	std::function< T(const Vector< T, D > &)>	sdf,
		const Vector< T, D > &	p,
		T	s,
		const Vector< T, D > &	center = (T(0)) )

Function to scale a geometry The object has to be placed in the origin of the coodinate system.

Parameters

sdf	signed Distance Function of the object which is altered
p	point for which the distance to the surface is calculated
s	scaling factor
center	vector containing the center, to shift the object to the origin of the coordinate system for the calculation of the elongation

Definition at line 437 of file sdf.h.

{
  return sdf((p - center) / s) * s;
}

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signedDistanceToPorosity()

template<typename T >

T olb::sdf::signedDistanceToPorosity	(	T	signedDist,
		T	eps )

Converts signed distance to values for the smooth epsilon boundary.

Definition at line 446 of file sdf.h.

{
  const T d = signedDist + .5 * eps;
  return util::pow(util::cos(M_PI2 * d / eps), 2);
}

References olb::util::cos(), M_PI2, and olb::util::pow().

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smooth_intersection()

template<typename T >

T olb::sdf::smooth_intersection	(	T	a,
		T	b,
		T	k )

Definition at line 361 of file sdf.h.

{
  T h = clamp(0.5 - 0.5 * (b - a) / k, 0.0, 1.0);
  return mix(b, a, h) + k * h * (1.0 - h);
}

References clamp(), and mix().

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smooth_subtraction()

template<typename T >

T olb::sdf::smooth_subtraction	(	T	a,
		T	b,
		T	k )

Definition at line 354 of file sdf.h.

{
  T h = clamp(0.5 - 0.5 * (b + a) / k, 0.0, 1.0);
  return mix(b, -a, h) + k * h * (1.0 - h);
}

References clamp(), and mix().

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smooth_union()

template<typename T >

T olb::sdf::smooth_union	(	T	a,
		T	b,
		T	k )

Definition at line 347 of file sdf.h.

{
  T h = clamp(0.5 + 0.5 * (b - a) / k, 0.0, 1.0);
  return mix(a, b, h) - k * h * (1.0 - h);
}

References clamp(), and mix().

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solidAngle()

template<typename T >

T olb::sdf::solidAngle	(	Vector< T, 3 >	p,
		Vector< T, 2 >	c,
		T	r )

Exact signed distance to the surface of a solid angle which represents a combination of a cone (axis y=0 to y=r) and a sphere.

Parameters

p	point for which the distance to the surface is calculated
c[0]	sin of the angle
c[1]	cos of the angle
r	radius of the sphere

Definition at line 286 of file sdf.h.

{
  Vector<T, 2> q {norm(Vector<T, 2> {p[0], p[2]}), p[1]};
  T            l = norm(q) - r;
  T            m = norm(q - c * clamp(q * c, 0.0, r));
  return util::max(l, m * util::sign(c[1] * q[0] - c[0] * q[1]));
}

References clamp(), olb::util::max(), olb::norm(), and olb::util::sign().

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sphere()

template<typename T , unsigned D>

T olb::sdf::sphere	(	Vector< T, D >	p,
		T	r )

Exact signed distance to the surface of circle (in 2D) or sphere (in 3D).

Parameters

p	point for which the distance to the surface is calculated
r	radius describing the geometry

Definition at line 104 of file sdf.h.

{
  return norm(p) - r;
}

References olb::norm().

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subtraction()

template<typename T >

T olb::sdf::subtraction	(	T	a,
		T	b )

Volume of a is subtracted from b.

The returned distance is not exact if the signed Distance of both objects points to the part of the surface which is removed for creating the combined object

Parameters

a	signed Distance of the first object
b	signed Distance of the second object

Definition at line 317 of file sdf.h.

{
  return util::max(-a, b);
}

References olb::util::max().

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torus()

template<typename T >

T olb::sdf::torus	(	Vector< T, 3 >	p,
		Vector< T, 2 >	t )

Exact signed distance to the surface of a torus placed in the XZ-plane of the coordinate system.

Parameters

p	point for which the distance to the surface is calculated
t[0]	distance center of the tube to center of the torus
t[1]	radius of the tube

Definition at line 271 of file sdf.h.

{
  Vector<T, 2> b {p[0], p[2]};
  Vector<T, 2> q {norm(b) - t[0], p[1]};
  return norm(q) - t[1];
}

References olb::norm().

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translate()

template<typename T , unsigned D>

Vector< T, D > olb::sdf::translate	(	Vector< T, D >	p,
		Vector< T, D >	origin )

Translation: The output of this function is used for the calculation of the signed distance to translated/shifted objects.

Parameters

p	point for which the distance to the surface is calculated
origin	point to which the object is shifted to

Definition at line 299 of file sdf.h.

{
  return p - origin;
}

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triangle()

template<typename T >

T olb::sdf::triangle	(	Vector< T, 2 >	p,
		Vector< T, 2 >	a,
		Vector< T, 2 >	b,
		Vector< T, 2 >	c )

Exact signed distance to the surface of two-dimensional triangle.

Parameters

p	point for which the distance to the surface is calculated
a	vector containing first vertex
b	vector containing second vertex
c	vector containing third vertex

Definition at line 143 of file sdf.h.

{
  Vector<T, 2> e0  = b - a;
  Vector<T, 2> v0  = p - a;
  Vector<T, 2> e1  = c - b;
  Vector<T, 2> v1  = p - b;
  Vector<T, 2> e2  = a - c;
  Vector<T, 2> v2  = p - c;
  Vector<T, 2> pq0 = v0 - e0 * clamp((v0 * e0) / (e0 * e0), 0., 1.);
  Vector<T, 2> pq1 = v1 - e1 * clamp((v1 * e1) / (e1 * e1), 0., 1.);
  Vector<T, 2> pq2 = v2 - e2 * clamp((v2 * e2) / (e2 * e2), 0., 1.);
  T            s   = util::sign(e0[0] * e2[1] - e0[1] * e2[0]);
 
  T dx = util::min(util::min((pq0 * pq0), (pq1 * pq1)), (pq2 * pq2));
  T dy = util::min(util::min(s * (v0[0] * e0[1] - v0[1] * e0[0]),
                             s * (v1[0] * e1[1] - v1[1] * e1[0])),
                   s * (v2[0] * e2[1] - v2[1] * e2[0]));
  return -util::sqrt(dx) * util::sign(dy);
}

References clamp(), olb::util::min(), olb::util::sign(), and olb::util::sqrt().

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unify()

template<typename T >

T olb::sdf::unify	(	T	a,
		T	b )

Volume of a and volume of b are combined as a new object.

The returned distance is not exact if the signed Distance of both objects points to the part of the surface which is removed for creating the combined object

Parameters

a	signed Distance of the first object
b	signed Distance of the second object

Definition at line 329 of file sdf.h.

{
  return util::min(a, b);
}

References olb::util::min().

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