Math for vkframes: the working precision, quaternions and 4x4 transforms.
Quaternions are q(4) = [x, y, z, w] (vector part first, as OpenFrames and
OpenSceneGraph store them), unit length for rotations. The rotation of q
takes vectors in a frame to its parent frame: v_parent = rotate(q, v_local).
Matrices are 4x4, column-major, acting on column vectors.
module~~vkf_math~~UsesGraph
module~vkf_math
vkf_math
iso_fortran_env
iso_fortran_env
module~vkf_math->iso_fortran_env
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modules which it uses.
module~~vkf_math~~UsedByGraph
module~vkf_math
vkf_math
module~vkf_artists
vkf_artists
module~vkf_artists->module~vkf_math
module~vkf_frames
vkf_frames
module~vkf_artists->module~vkf_frames
module~vkf_line_chain
vkf_line_chain
module~vkf_artists->module~vkf_line_chain
module~vkf_pick
vkf_pick
module~vkf_artists->module~vkf_pick
module~vkf_trajectory
vkf_trajectory
module~vkf_artists->module~vkf_trajectory
module~vkf_clock
vkf_clock
module~vkf_clock->module~vkf_math
module~vkf_diagnostics
vkf_diagnostics
module~vkf_diagnostics->module~vkf_math
module~vkf_follower
vkf_follower
module~vkf_follower->module~vkf_math
module~vkf_follower->module~vkf_trajectory
module~vkf_frames->module~vkf_math
module~vkf_frames->module~vkf_follower
module~vkf_geometry
vkf_geometry
module~vkf_frames->module~vkf_geometry
module~vkf_frames->module~vkf_pick
module~vkf_text
vkf_text
module~vkf_frames->module~vkf_text
module~vkf_frames->module~vkf_trajectory
module~vkf_geometry->module~vkf_math
module~vkf_gui
vkf_gui
module~vkf_gui->module~vkf_math
module~vkf_handles
vkf_handles
module~vkf_handles->module~vkf_math
module~vkf_handles->module~vkf_artists
module~vkf_handles->module~vkf_frames
module~vkf_handles->module~vkf_geometry
module~vkf_handles->module~vkf_pick
module~vkf_handles->module~vkf_trajectory
module~vkf_hud
vkf_hud
module~vkf_hud->module~vkf_math
module~vkf_hud->module~vkf_frames
module~vkf_line_chain->module~vkf_math
module~vkf_line_segments
vkf_line_segments
module~vkf_line_segments->module~vkf_math
module~vkf_line_segments->module~vkf_frames
module~vkf_line_segments->module~vkf_line_chain
module~vkf_model
vkf_model
module~vkf_model->module~vkf_math
module~vkf_model->module~vkf_frames
module~vkf_model->module~vkf_pick
module~vkf_pick->module~vkf_math
module~vkf_shadow
vkf_shadow
module~vkf_shadow->module~vkf_math
module~vkf_shadow->module~vkf_frames
module~vkf_sphere
vkf_sphere
module~vkf_shadow->module~vkf_sphere
module~vkf_terrain
vkf_terrain
module~vkf_shadow->module~vkf_terrain
module~vkf_shapes
vkf_shapes
module~vkf_shapes->module~vkf_math
module~vkf_shapes->module~vkf_artists
module~vkf_shapes->module~vkf_frames
module~vkf_sky
vkf_sky
module~vkf_sky->module~vkf_math
module~vkf_sky->module~vkf_text
module~vkf_sphere->module~vkf_math
module~vkf_sphere->module~vkf_frames
module~vkf_sphere->module~vkf_geometry
module~vkf_sphere->module~vkf_pick
module~vkf_terrain->module~vkf_math
module~vkf_terrain->module~vkf_frames
module~vkf_terrain->module~vkf_pick
module~vkf_text->module~vkf_math
module~vkf_trajectory->module~vkf_math
module~vkf_view
vkf_view
module~vkf_view->module~vkf_math
module~vkf_view->module~vkf_frames
module~vkf_window
vkf_window
module~vkf_window->module~vkf_math
module~vkf_window->module~vkf_clock
module~vkf_window->module~vkf_diagnostics
module~vkf_window->module~vkf_frames
module~vkf_window->module~vkf_gui
module~vkf_window->module~vkf_handles
module~vkf_window->module~vkf_hud
module~vkf_window->module~vkf_pick
module~vkf_window->module~vkf_sky
module~vkf_window->module~vkf_text
module~vkf_window->module~vkf_view
module~vkframes
vkframes
module~vkframes->module~vkf_math
module~vkframes->module~vkf_artists
module~vkframes->module~vkf_clock
module~vkframes->module~vkf_diagnostics
module~vkframes->module~vkf_follower
module~vkframes->module~vkf_frames
module~vkframes->module~vkf_gui
module~vkframes->module~vkf_handles
module~vkframes->module~vkf_hud
module~vkframes->module~vkf_line_segments
module~vkframes->module~vkf_model
module~vkframes->module~vkf_pick
module~vkframes->module~vkf_shadow
module~vkframes->module~vkf_shapes
module~vkframes->module~vkf_sky
module~vkframes->module~vkf_sphere
module~vkframes->module~vkf_terrain
module~vkframes->module~vkf_trajectory
module~vkframes->module~vkf_view
module~vkframes->module~vkf_window
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Variables
Type
Visibility Attributes
Name
Initial
public,
parameter
::
pi =
acos(-1.0_wp)
pi
public,
parameter
::
quat_identity (4) =
[0.0_wp, 0.0_wp, 0.0_wp, 1.0_wp]
the identity rotation
public,
parameter
::
wp =
real64
working precision
Functions
The cross product a x b.
Arguments
Type
Intent Optional Attributes
Name
intent(in)
::
a (3)
the vectors
intent(in)
::
b (3)
the vectors
Return Value
real(kind=wp), (3)
A frame's local-to-parent transform, as OpenFrames' FrameTransform builds
it: translate(position) * rotate(attitude) * scale(scale) * translate(-pivot).
Arguments
Type
Intent Optional Attributes
Name
intent(in)
::
position (3)
origin in the parent frame
intent(in)
::
attitude (4)
quaternion [x, y, z, w]
intent(in),
optional
::
scale (3)
scale along each local axis (default 1)
intent(in),
optional
::
pivot (3)
local point placed at position (default 0)
Return Value
real(kind=wp), (4,4)
The 4x4 identity matrix.
Arguments
None
Return Value
real(kind=wp), (4,4)
The conjugate of q: the inverse rotation, for a unit quaternion.
Arguments
Type
Intent Optional Attributes
Name
intent(in)
::
q (4)
quaternion [x, y, z, w]
Return Value
real(kind=wp), (4)
The rotation by angle (radians) about axis (need not be unit length).
Arguments
Type
Intent Optional Attributes
Name
intent(in)
::
axis (3)
rotation axis (any length; zero: the identity)
intent(in)
::
angle
(radians)
Return Value
real(kind=wp), (4)
The unit quaternion of a 3x3 rotation matrix (columns: the rotated
axes; the inverse of quat_to_matrix), by Shepperd's method (from the
largest of w and the vector components, for accuracy).
Arguments
Type
Intent Optional Attributes
Name
intent(in)
::
m (3,3)
rotation matrix (columns: the rotated axes)
Return Value
real(kind=wp), (4)
The shortest rotation taking the direction of from to that of to
(OpenSceneGraph's osg::Quat::makeRotate(from, to)); a half turn about a
perpendicular axis if they are opposite, the identity if either is zero.
Arguments
Type
Intent Optional Attributes
Name
intent(in)
::
from (3)
directions (any length)
intent(in)
::
to (3)
directions (any length)
Return Value
real(kind=wp), (4)
The product a*b: rotating by b, then by a.
Arguments
Type
Intent Optional Attributes
Name
intent(in)
::
a (4)
quaternions [x, y, z, w]
intent(in)
::
b (4)
quaternions [x, y, z, w]
Return Value
real(kind=wp), (4)
q / |q| (the identity for a zero quaternion).
Arguments
Type
Intent Optional Attributes
Name
intent(in)
::
q (4)
quaternion [x, y, z, w] (any length)
Return Value
real(kind=wp), (4)
The vector v rotated by the unit quaternion q.
Arguments
Type
Intent Optional Attributes
Name
intent(in)
::
q (4)
unit quaternion [x, y, z, w], and the vector
intent(in)
::
v (3)
unit quaternion [x, y, z, w], and the vector
Return Value
real(kind=wp), (3)
Spherical linear interpolation from a (t = 0) to b (t = 1), along the
shorter arc; linear when the two are nearly equal. As OpenSceneGraph's
osg::Quat::slerp, which OpenFrames' TrajectoryFollower uses, except that
the linear case is normalized (OSG's isn't: a quaternion a little short of
unit length, its rotation matrix then also scaling, by a millionth for
attitudes 0.25 deg apart, a kilometre at the Moon's distance).
Arguments
Type
Intent Optional Attributes
Name
intent(in)
::
t
the fraction of the way (0 to 1)
intent(in)
::
a (4)
unit quaternions [x, y, z, w] at t = 0 and t = 1
intent(in)
::
b (4)
unit quaternions [x, y, z, w] at t = 0 and t = 1
Return Value
real(kind=wp), (4)
The 3x3 rotation matrix of the unit quaternion q (columns: the rotated axes).
Arguments
Type
Intent Optional Attributes
Name
intent(in)
::
q (4)
unit quaternion [x, y, z, w]
Return Value
real(kind=wp), (3,3)
The inverse of a rigid transform (rotation and translation): [R' -R't].
Arguments
Type
Intent Optional Attributes
Name
intent(in)
::
m (4,4)
rigid transform (rotation and translation)
Return Value
real(kind=wp), (4,4)
The 4x4 transform translating by t.
Arguments
Type
Intent Optional Attributes
Name
intent(in)
::
t (3)
translation
Return Value
real(kind=wp), (4,4)
Subroutines
Set rgba from color: [r, g, b] (alpha unchanged) or [r, g, b, a].
Arguments
Type
Intent Optional Attributes
Name
intent(inout)
::
rgba (4)
[r, g, b, a]
intent(in)
::
color (:)
[r, g, b] or [r, g, b, a] (0 to 1)