orbit-plus/include/cglm/quat.h

868 lines
20 KiB
C
Raw Permalink Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

/*
* Copyright (c), Recep Aslantas.
*
* MIT License (MIT), http://opensource.org/licenses/MIT
* Full license can be found in the LICENSE file
*/
/*
Macros:
GLM_QUAT_IDENTITY_INIT
GLM_QUAT_IDENTITY
Functions:
CGLM_INLINE void glm_quat_identity(versor q);
CGLM_INLINE void glm_quat_init(versor q, float x, float y, float z, float w);
CGLM_INLINE void glm_quat(versor q, float angle, float x, float y, float z);
CGLM_INLINE void glm_quatv(versor q, float angle, vec3 axis);
CGLM_INLINE void glm_quat_copy(versor q, versor dest);
CGLM_INLINE void glm_quat_from_vecs(vec3 a, vec3 b, versor dest);
CGLM_INLINE float glm_quat_norm(versor q);
CGLM_INLINE void glm_quat_normalize(versor q);
CGLM_INLINE void glm_quat_normalize_to(versor q, versor dest);
CGLM_INLINE float glm_quat_dot(versor p, versor q);
CGLM_INLINE void glm_quat_conjugate(versor q, versor dest);
CGLM_INLINE void glm_quat_inv(versor q, versor dest);
CGLM_INLINE void glm_quat_add(versor p, versor q, versor dest);
CGLM_INLINE void glm_quat_sub(versor p, versor q, versor dest);
CGLM_INLINE float glm_quat_real(versor q);
CGLM_INLINE void glm_quat_imag(versor q, vec3 dest);
CGLM_INLINE void glm_quat_imagn(versor q, vec3 dest);
CGLM_INLINE float glm_quat_imaglen(versor q);
CGLM_INLINE float glm_quat_angle(versor q);
CGLM_INLINE void glm_quat_axis(versor q, vec3 dest);
CGLM_INLINE void glm_quat_mul(versor p, versor q, versor dest);
CGLM_INLINE void glm_quat_mat4(versor q, mat4 dest);
CGLM_INLINE void glm_quat_mat4t(versor q, mat4 dest);
CGLM_INLINE void glm_quat_mat3(versor q, mat3 dest);
CGLM_INLINE void glm_quat_mat3t(versor q, mat3 dest);
CGLM_INLINE void glm_quat_lerp(versor from, versor to, float t, versor dest);
CGLM_INLINE void glm_quat_lerpc(versor from, versor to, float t, versor dest);
CGLM_INLINE void glm_quat_slerp(versor q, versor r, float t, versor dest);
CGLM_INLINE void glm_quat_nlerp(versor q, versor r, float t, versor dest);
CGLM_INLINE void glm_quat_look(vec3 eye, versor ori, mat4 dest);
CGLM_INLINE void glm_quat_for(vec3 dir, vec3 fwd, vec3 up, versor dest);
CGLM_INLINE void glm_quat_forp(vec3 from,
vec3 to,
vec3 fwd,
vec3 up,
versor dest);
CGLM_INLINE void glm_quat_rotatev(versor q, vec3 v, vec3 dest);
CGLM_INLINE void glm_quat_rotate(mat4 m, versor q, mat4 dest);
*/
#ifndef cglm_quat_h
#define cglm_quat_h
#include "common.h"
#include "vec3.h"
#include "vec4.h"
#include "mat4.h"
#include "mat3.h"
#include "affine-mat.h"
#include "affine.h"
#ifdef CGLM_SSE_FP
# include "simd/sse2/quat.h"
#endif
#ifdef CGLM_NEON_FP
# include "simd/neon/quat.h"
#endif
CGLM_INLINE void glm_quat_normalize(versor q);
/*
* IMPORTANT:
* ----------------------------------------------------------------------------
* cglm stores quat as [x, y, z, w] since v0.3.6
*
* it was [w, x, y, z] before v0.3.6 it has been changed to [x, y, z, w]
* with v0.3.6 version.
* ----------------------------------------------------------------------------
*/
#define GLM_QUAT_IDENTITY_INIT {0.0f, 0.0f, 0.0f, 1.0f}
#define GLM_QUAT_IDENTITY ((versor)GLM_QUAT_IDENTITY_INIT)
/*!
* @brief makes given quat to identity
*
* @param[in, out] q quaternion
*/
CGLM_INLINE
void
glm_quat_identity(versor q) {
CGLM_ALIGN(16) versor v = GLM_QUAT_IDENTITY_INIT;
glm_vec4_copy(v, q);
}
/*!
* @brief make given quaternion array's each element identity quaternion
*
* @param[in, out] q quat array (must be aligned (16)
* if alignment is not disabled)
*
* @param[in] count count of quaternions
*/
CGLM_INLINE
void
glm_quat_identity_array(versor * __restrict q, size_t count) {
CGLM_ALIGN(16) versor v = GLM_QUAT_IDENTITY_INIT;
size_t i;
for (i = 0; i < count; i++) {
glm_vec4_copy(v, q[i]);
}
}
/*!
* @brief inits quaterion with raw values
*
* @param[out] q quaternion
* @param[in] x x
* @param[in] y y
* @param[in] z z
* @param[in] w w (real part)
*/
CGLM_INLINE
void
glm_quat_init(versor q, float x, float y, float z, float w) {
q[0] = x;
q[1] = y;
q[2] = z;
q[3] = w;
}
/*!
* @brief creates NEW quaternion with axis vector
*
* @param[out] q quaternion
* @param[in] angle angle (radians)
* @param[in] axis axis
*/
CGLM_INLINE
void
glm_quatv(versor q, float angle, vec3 axis) {
CGLM_ALIGN(8) vec3 k;
float a, c, s;
a = angle * 0.5f;
c = cosf(a);
s = sinf(a);
glm_normalize_to(axis, k);
q[0] = s * k[0];
q[1] = s * k[1];
q[2] = s * k[2];
q[3] = c;
}
/*!
* @brief creates NEW quaternion with individual axis components
*
* @param[out] q quaternion
* @param[in] angle angle (radians)
* @param[in] x axis.x
* @param[in] y axis.y
* @param[in] z axis.z
*/
CGLM_INLINE
void
glm_quat(versor q, float angle, float x, float y, float z) {
CGLM_ALIGN(8) vec3 axis = {x, y, z};
glm_quatv(q, angle, axis);
}
/*!
* @brief copy quaternion to another one
*
* @param[in] q quaternion
* @param[out] dest destination
*/
CGLM_INLINE
void
glm_quat_copy(versor q, versor dest) {
glm_vec4_copy(q, dest);
}
/*!
* @brief compute quaternion rotating vector A to vector B
*
* @param[in] a vec3 (must have unit length)
* @param[in] b vec3 (must have unit length)
* @param[out] dest quaternion (of unit length)
*/
CGLM_INLINE
void
glm_quat_from_vecs(vec3 a, vec3 b, versor dest) {
CGLM_ALIGN(8) vec3 axis;
float cos_theta;
float cos_half_theta;
cos_theta = glm_vec3_dot(a, b);
if (cos_theta >= 1.f - GLM_FLT_EPSILON) { /* a ∥ b */
glm_quat_identity(dest);
return;
}
if (cos_theta < -1.f + GLM_FLT_EPSILON) { /* angle(a, b) = π */
glm_vec3_ortho(a, axis);
cos_half_theta = 0.f; /* cos π/2 */
} else {
glm_vec3_cross(a, b, axis);
cos_half_theta = 1.0f + cos_theta; /* cos 0 + cos θ */
}
glm_quat_init(dest, axis[0], axis[1], axis[2], cos_half_theta);
glm_quat_normalize(dest);
}
/*!
* @brief returns norm (magnitude) of quaternion
*
* @param[in] q quaternion
*/
CGLM_INLINE
float
glm_quat_norm(versor q) {
return glm_vec4_norm(q);
}
/*!
* @brief normalize quaternion and store result in dest
*
* @param[in] q quaternion to normalze
* @param[out] dest destination quaternion
*/
CGLM_INLINE
void
glm_quat_normalize_to(versor q, versor dest) {
#if defined( __SSE2__ ) || defined( __SSE2__ )
__m128 xdot, x0;
float dot;
x0 = glmm_load(q);
xdot = glmm_vdot(x0, x0);
dot = _mm_cvtss_f32(xdot);
if (dot <= 0.0f) {
glm_quat_identity(dest);
return;
}
glmm_store(dest, _mm_div_ps(x0, _mm_sqrt_ps(xdot)));
#else
float dot;
dot = glm_vec4_norm2(q);
if (dot <= 0.0f) {
glm_quat_identity(dest);
return;
}
glm_vec4_scale(q, 1.0f / sqrtf(dot), dest);
#endif
}
/*!
* @brief normalize quaternion
*
* @param[in, out] q quaternion
*/
CGLM_INLINE
void
glm_quat_normalize(versor q) {
glm_quat_normalize_to(q, q);
}
/*!
* @brief dot product of two quaternion
*
* @param[in] p quaternion 1
* @param[in] q quaternion 2
*/
CGLM_INLINE
float
glm_quat_dot(versor p, versor q) {
return glm_vec4_dot(p, q);
}
/*!
* @brief conjugate of quaternion
*
* @param[in] q quaternion
* @param[out] dest conjugate
*/
CGLM_INLINE
void
glm_quat_conjugate(versor q, versor dest) {
glm_vec4_negate_to(q, dest);
dest[3] = -dest[3];
}
/*!
* @brief inverse of non-zero quaternion
*
* @param[in] q quaternion
* @param[out] dest inverse quaternion
*/
CGLM_INLINE
void
glm_quat_inv(versor q, versor dest) {
CGLM_ALIGN(16) versor conj;
glm_quat_conjugate(q, conj);
glm_vec4_scale(conj, 1.0f / glm_vec4_norm2(q), dest);
}
/*!
* @brief add (componentwise) two quaternions and store result in dest
*
* @param[in] p quaternion 1
* @param[in] q quaternion 2
* @param[out] dest result quaternion
*/
CGLM_INLINE
void
glm_quat_add(versor p, versor q, versor dest) {
glm_vec4_add(p, q, dest);
}
/*!
* @brief subtract (componentwise) two quaternions and store result in dest
*
* @param[in] p quaternion 1
* @param[in] q quaternion 2
* @param[out] dest result quaternion
*/
CGLM_INLINE
void
glm_quat_sub(versor p, versor q, versor dest) {
glm_vec4_sub(p, q, dest);
}
/*!
* @brief returns real part of quaternion
*
* @param[in] q quaternion
*/
CGLM_INLINE
float
glm_quat_real(versor q) {
return q[3];
}
/*!
* @brief returns imaginary part of quaternion
*
* @param[in] q quaternion
* @param[out] dest imag
*/
CGLM_INLINE
void
glm_quat_imag(versor q, vec3 dest) {
dest[0] = q[0];
dest[1] = q[1];
dest[2] = q[2];
}
/*!
* @brief returns normalized imaginary part of quaternion
*
* @param[in] q quaternion
*/
CGLM_INLINE
void
glm_quat_imagn(versor q, vec3 dest) {
glm_normalize_to(q, dest);
}
/*!
* @brief returns length of imaginary part of quaternion
*
* @param[in] q quaternion
*/
CGLM_INLINE
float
glm_quat_imaglen(versor q) {
return glm_vec3_norm(q);
}
/*!
* @brief returns angle of quaternion
*
* @param[in] q quaternion
*/
CGLM_INLINE
float
glm_quat_angle(versor q) {
/*
sin(theta / 2) = length(x*x + y*y + z*z)
cos(theta / 2) = w
theta = 2 * atan(sin(theta / 2) / cos(theta / 2))
*/
return 2.0f * atan2f(glm_quat_imaglen(q), glm_quat_real(q));
}
/*!
* @brief axis of quaternion
*
* @param[in] q quaternion
* @param[out] dest axis of quaternion
*/
CGLM_INLINE
void
glm_quat_axis(versor q, vec3 dest) {
glm_quat_imagn(q, dest);
}
/*!
* @brief multiplies two quaternion and stores result in dest
* this is also called Hamilton Product
*
* According to WikiPedia:
* The product of two rotation quaternions [clarification needed] will be
* equivalent to the rotation q followed by the rotation p
*
* @param[in] p quaternion 1
* @param[in] q quaternion 2
* @param[out] dest result quaternion
*/
CGLM_INLINE
void
glm_quat_mul(versor p, versor q, versor dest) {
/*
+ (a1 b2 + b1 a2 + c1 d2 d1 c2)i
+ (a1 c2 b1 d2 + c1 a2 + d1 b2)j
+ (a1 d2 + b1 c2 c1 b2 + d1 a2)k
a1 a2 b1 b2 c1 c2 d1 d2
*/
#if defined( __SSE__ ) || defined( __SSE2__ )
glm_quat_mul_sse2(p, q, dest);
#elif defined(CGLM_NEON_FP)
glm_quat_mul_neon(p, q, dest);
#else
dest[0] = p[3] * q[0] + p[0] * q[3] + p[1] * q[2] - p[2] * q[1];
dest[1] = p[3] * q[1] - p[0] * q[2] + p[1] * q[3] + p[2] * q[0];
dest[2] = p[3] * q[2] + p[0] * q[1] - p[1] * q[0] + p[2] * q[3];
dest[3] = p[3] * q[3] - p[0] * q[0] - p[1] * q[1] - p[2] * q[2];
#endif
}
/*!
* @brief convert quaternion to mat4
*
* @param[in] q quaternion
* @param[out] dest result matrix
*/
CGLM_INLINE
void
glm_quat_mat4(versor q, mat4 dest) {
float w, x, y, z,
xx, yy, zz,
xy, yz, xz,
wx, wy, wz, norm, s;
norm = glm_quat_norm(q);
s = norm > 0.0f ? 2.0f / norm : 0.0f;
x = q[0];
y = q[1];
z = q[2];
w = q[3];
xx = s * x * x; xy = s * x * y; wx = s * w * x;
yy = s * y * y; yz = s * y * z; wy = s * w * y;
zz = s * z * z; xz = s * x * z; wz = s * w * z;
dest[0][0] = 1.0f - yy - zz;
dest[1][1] = 1.0f - xx - zz;
dest[2][2] = 1.0f - xx - yy;
dest[0][1] = xy + wz;
dest[1][2] = yz + wx;
dest[2][0] = xz + wy;
dest[1][0] = xy - wz;
dest[2][1] = yz - wx;
dest[0][2] = xz - wy;
dest[0][3] = 0.0f;
dest[1][3] = 0.0f;
dest[2][3] = 0.0f;
dest[3][0] = 0.0f;
dest[3][1] = 0.0f;
dest[3][2] = 0.0f;
dest[3][3] = 1.0f;
}
/*!
* @brief convert quaternion to mat4 (transposed)
*
* @param[in] q quaternion
* @param[out] dest result matrix as transposed
*/
CGLM_INLINE
void
glm_quat_mat4t(versor q, mat4 dest) {
float w, x, y, z,
xx, yy, zz,
xy, yz, xz,
wx, wy, wz, norm, s;
norm = glm_quat_norm(q);
s = norm > 0.0f ? 2.0f / norm : 0.0f;
x = q[0];
y = q[1];
z = q[2];
w = q[3];
xx = s * x * x; xy = s * x * y; wx = s * w * x;
yy = s * y * y; yz = s * y * z; wy = s * w * y;
zz = s * z * z; xz = s * x * z; wz = s * w * z;
dest[0][0] = 1.0f - yy - zz;
dest[1][1] = 1.0f - xx - zz;
dest[2][2] = 1.0f - xx - yy;
dest[1][0] = xy + wz;
dest[2][1] = yz + wx;
dest[0][2] = xz + wy;
dest[0][1] = xy - wz;
dest[1][2] = yz - wx;
dest[2][0] = xz - wy;
dest[0][3] = 0.0f;
dest[1][3] = 0.0f;
dest[2][3] = 0.0f;
dest[3][0] = 0.0f;
dest[3][1] = 0.0f;
dest[3][2] = 0.0f;
dest[3][3] = 1.0f;
}
/*!
* @brief convert quaternion to mat3
*
* @param[in] q quaternion
* @param[out] dest result matrix
*/
CGLM_INLINE
void
glm_quat_mat3(versor q, mat3 dest) {
float w, x, y, z,
xx, yy, zz,
xy, yz, xz,
wx, wy, wz, norm, s;
norm = glm_quat_norm(q);
s = norm > 0.0f ? 2.0f / norm : 0.0f;
x = q[0];
y = q[1];
z = q[2];
w = q[3];
xx = s * x * x; xy = s * x * y; wx = s * w * x;
yy = s * y * y; yz = s * y * z; wy = s * w * y;
zz = s * z * z; xz = s * x * z; wz = s * w * z;
dest[0][0] = 1.0f - yy - zz;
dest[1][1] = 1.0f - xx - zz;
dest[2][2] = 1.0f - xx - yy;
dest[0][1] = xy + wz;
dest[1][2] = yz + wx;
dest[2][0] = xz + wy;
dest[1][0] = xy - wz;
dest[2][1] = yz - wx;
dest[0][2] = xz - wy;
}
/*!
* @brief convert quaternion to mat3 (transposed)
*
* @param[in] q quaternion
* @param[out] dest result matrix
*/
CGLM_INLINE
void
glm_quat_mat3t(versor q, mat3 dest) {
float w, x, y, z,
xx, yy, zz,
xy, yz, xz,
wx, wy, wz, norm, s;
norm = glm_quat_norm(q);
s = norm > 0.0f ? 2.0f / norm : 0.0f;
x = q[0];
y = q[1];
z = q[2];
w = q[3];
xx = s * x * x; xy = s * x * y; wx = s * w * x;
yy = s * y * y; yz = s * y * z; wy = s * w * y;
zz = s * z * z; xz = s * x * z; wz = s * w * z;
dest[0][0] = 1.0f - yy - zz;
dest[1][1] = 1.0f - xx - zz;
dest[2][2] = 1.0f - xx - yy;
dest[1][0] = xy + wz;
dest[2][1] = yz + wx;
dest[0][2] = xz + wy;
dest[0][1] = xy - wz;
dest[1][2] = yz - wx;
dest[2][0] = xz - wy;
}
/*!
* @brief interpolates between two quaternions
* using linear interpolation (LERP)
*
* @param[in] from from
* @param[in] to to
* @param[in] t interpolant (amount)
* @param[out] dest result quaternion
*/
CGLM_INLINE
void
glm_quat_lerp(versor from, versor to, float t, versor dest) {
glm_vec4_lerp(from, to, t, dest);
}
/*!
* @brief interpolates between two quaternions
* using linear interpolation (LERP)
*
* @param[in] from from
* @param[in] to to
* @param[in] t interpolant (amount) clamped between 0 and 1
* @param[out] dest result quaternion
*/
CGLM_INLINE
void
glm_quat_lerpc(versor from, versor to, float t, versor dest) {
glm_vec4_lerpc(from, to, t, dest);
}
/*!
* @brief interpolates between two quaternions
* taking the shortest rotation path using
* normalized linear interpolation (NLERP)
*
* @param[in] from from
* @param[in] to to
* @param[in] t interpolant (amount)
* @param[out] dest result quaternion
*/
CGLM_INLINE
void
glm_quat_nlerp(versor from, versor to, float t, versor dest) {
versor target;
float dot;
dot = glm_vec4_dot(from, to);
glm_vec4_scale(to, (dot >= 0) ? 1.0f : -1.0f, target);
glm_quat_lerp(from, target, t, dest);
glm_quat_normalize(dest);
}
/*!
* @brief interpolates between two quaternions
* using spherical linear interpolation (SLERP)
*
* @param[in] from from
* @param[in] to to
* @param[in] t amout
* @param[out] dest result quaternion
*/
CGLM_INLINE
void
glm_quat_slerp(versor from, versor to, float t, versor dest) {
CGLM_ALIGN(16) vec4 q1, q2;
float cosTheta, sinTheta, angle;
cosTheta = glm_quat_dot(from, to);
glm_quat_copy(from, q1);
if (fabsf(cosTheta) >= 1.0f) {
glm_quat_copy(q1, dest);
return;
}
if (cosTheta < 0.0f) {
glm_vec4_negate(q1);
cosTheta = -cosTheta;
}
sinTheta = sqrtf(1.0f - cosTheta * cosTheta);
/* LERP to avoid zero division */
if (fabsf(sinTheta) < 0.001f) {
glm_quat_lerp(from, to, t, dest);
return;
}
/* SLERP */
angle = acosf(cosTheta);
glm_vec4_scale(q1, sinf((1.0f - t) * angle), q1);
glm_vec4_scale(to, sinf(t * angle), q2);
glm_vec4_add(q1, q2, q1);
glm_vec4_scale(q1, 1.0f / sinTheta, dest);
}
/*!
* @brief creates view matrix using quaternion as camera orientation
*
* @param[in] eye eye
* @param[in] ori orientation in world space as quaternion
* @param[out] dest view matrix
*/
CGLM_INLINE
void
glm_quat_look(vec3 eye, versor ori, mat4 dest) {
/* orientation */
glm_quat_mat4t(ori, dest);
/* translate */
glm_mat4_mulv3(dest, eye, 1.0f, dest[3]);
glm_vec3_negate(dest[3]);
}
/*!
* @brief creates look rotation quaternion
*
* @param[in] dir direction to look
* @param[in] up up vector
* @param[out] dest destination quaternion
*/
CGLM_INLINE
void
glm_quat_for(vec3 dir, vec3 up, versor dest) {
CGLM_ALIGN_MAT mat3 m;
glm_vec3_normalize_to(dir, m[2]);
/* No need to negate in LH, but we use RH here */
glm_vec3_negate(m[2]);
glm_vec3_crossn(up, m[2], m[0]);
glm_vec3_cross(m[2], m[0], m[1]);
glm_mat3_quat(m, dest);
}
/*!
* @brief creates look rotation quaternion using source and
* destination positions p suffix stands for position
*
* @param[in] from source point
* @param[in] to destination point
* @param[in] up up vector
* @param[out] dest destination quaternion
*/
CGLM_INLINE
void
glm_quat_forp(vec3 from, vec3 to, vec3 up, versor dest) {
CGLM_ALIGN(8) vec3 dir;
glm_vec3_sub(to, from, dir);
glm_quat_for(dir, up, dest);
}
/*!
* @brief rotate vector using using quaternion
*
* @param[in] q quaternion
* @param[in] v vector to rotate
* @param[out] dest rotated vector
*/
CGLM_INLINE
void
glm_quat_rotatev(versor q, vec3 v, vec3 dest) {
CGLM_ALIGN(16) versor p;
CGLM_ALIGN(8) vec3 u, v1, v2;
float s;
glm_quat_normalize_to(q, p);
glm_quat_imag(p, u);
s = glm_quat_real(p);
glm_vec3_scale(u, 2.0f * glm_vec3_dot(u, v), v1);
glm_vec3_scale(v, s * s - glm_vec3_dot(u, u), v2);
glm_vec3_add(v1, v2, v1);
glm_vec3_cross(u, v, v2);
glm_vec3_scale(v2, 2.0f * s, v2);
glm_vec3_add(v1, v2, dest);
}
/*!
* @brief rotate existing transform matrix using quaternion
*
* @param[in] m existing transform matrix
* @param[in] q quaternion
* @param[out] dest rotated matrix/transform
*/
CGLM_INLINE
void
glm_quat_rotate(mat4 m, versor q, mat4 dest) {
CGLM_ALIGN_MAT mat4 rot;
glm_quat_mat4(q, rot);
glm_mul_rot(m, rot, dest);
}
/*!
* @brief rotate existing transform matrix using quaternion at pivot point
*
* @param[in, out] m existing transform matrix
* @param[in] q quaternion
* @param[out] pivot pivot
*/
CGLM_INLINE
void
glm_quat_rotate_at(mat4 m, versor q, vec3 pivot) {
CGLM_ALIGN(8) vec3 pivotInv;
glm_vec3_negate_to(pivot, pivotInv);
glm_translate(m, pivot);
glm_quat_rotate(m, q, m);
glm_translate(m, pivotInv);
}
/*!
* @brief rotate NEW transform matrix using quaternion at pivot point
*
* this creates rotation matrix, it assumes you don't have a matrix
*
* this should work faster than glm_quat_rotate_at because it reduces
* one glm_translate.
*
* @param[out] m existing transform matrix
* @param[in] q quaternion
* @param[in] pivot pivot
*/
CGLM_INLINE
void
glm_quat_rotate_atm(mat4 m, versor q, vec3 pivot) {
CGLM_ALIGN(8) vec3 pivotInv;
glm_vec3_negate_to(pivot, pivotInv);
glm_translate_make(m, pivot);
glm_quat_rotate(m, q, m);
glm_translate(m, pivotInv);
}
#endif /* cglm_quat_h */