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Transformation steps Translation = shifting Scaling Rotation Transvection = shear mapping Planar transformations (2D) Spatial transformations (3D) Rotate cuboid about centre axis In the library

Points in the plane and in 3D space are transformed by given parameters. A sequence of up to 13 individual transformation steps can be performed. In this way, all conceivable transformations can be configured.

≡ START First steps Deutsch

IN DUBIO PRO GEO computes planar and spatial coordinate transformations, i.e. conversions of point coordinates of an source system to coordinates of a target system. We do not require any special approximate alignment of the axes of the systems. Coordinates are specified via .

Cartesian lefthanded, cartesian righthanded systems (XYZ) and grid systems (northing,easting,height) (or other succession of coordinates) can by transformed immediately. Ellipsoidal systems (latitude,longitude,height) must first be converted with . See coordinate system type.

The coordinates in the source system are always denoted by x,y,z and in the target system by X,Y,Z . For grid coordinates we identify north = x and/or X, east = y and/or Y and height = z and/or Z.

≡ START First steps Deutsch Transformation steps

Each transformation can be imagined as a sequence of elementary transformation steps. By arbitrary combination of the steps

• Translation = shifting
• Scaling = scale change, for grid systems not including the grid scale factor
• Rotation
• Transvection = shear mapping
• Reflexion = mirroring from lefthanded to righthanded system or vice versa

using the parameters

shift and rotation parameters		scaling and shear parameters
tx	translation along x axis	mx	scale factor of x axis
ty	translation along y axis	my	scale factor of y axis
tz	translation along z axis	mz	scale factor of z axis
		mxy	scale factor of x and y axis
εx	Eulerian angle for rotation about x axis	m	scale factor of all axes
εy	Eulerian angle for rotation about x axis	fxy	shear factor for y vs. x axis
εz	Eulerian angle for rotation about z axis	fyx	shear factor for x vs. y axis
ε	2D rotation angle (identical to εz)	fxz	shear factor for z vs. x axis
	rotation angle about Eulerian axis	fzx	shear factor for x vs. z axis
ex	Eulerian axis, x vector component	fyz	shear factor for z vs. y axis
ey	Eulerian axis, y vector component	fzy	shear factor for y vs. z axis
ez	Eulerian axis, z vector component	τxy	shear angle for y vs. x axis
		τyx	shear angle for x vs. y axis
q0	quaternion, 0th component	τxz	shear angle for z vs. x axis
q1	quaternion, 1st component	τzx	shear angle for x vs. z axis
q2	quaternion, 2nd component	τyz	shear angle for z vs. y axis
q3	quaternion, 3rd component	τzy	shear angle for y vs. z axis

all relevant transformation types can be represented. In total up to 13 elementary steps can be arbitrarily combined. They are processed in the given succession. E.g., it is possible to perform a translation and a rotation followed by a further translation. The reflexion is automatically invoked, if the source system is a lefthanded system and the target system is a righthanded system or vice versa. In many cases the transformation result depends on the succession of the transformation steps.

The transformation steps involving the z coordinate require, that all points have three coordinates.

≡ START First steps Deutsch Translation = shifting

For planar transformations two translation parameters t_x , t_y, for spatial transformations three translation parameters t_x , t_y , t_z can be given. They are expected in the natural length unit , i.e. for grid systems not in grid scale. However, if this is desired differently, then grid systems must be redeclared to cartesian systems, see Bag of tricks. Missing translation parameters are treated as zero.

X Y

=

tx ty

+

x y

X Y Z

=

tx ty tz

+

x y z

≡ START First steps Deutsch Scaling

Scale parameters are scale factors applied to the coordinates. For planar transformations at most two scale parameters m_x , m_y , for spatial transformations at most three scale parameters m_x , m_y , m_z are accepted. All scale parameters must be positive. They do not include the grid scale factors . However, if this is desired differently, then grid systems must be redeclared to cartesian systems, see Bag of tricks. Missing scale parameters are treated as one.

X Y

=

mx·x my·y

X Y

= m·

x y

X Y Z

=

mx·x my·y mz·z

X Y Z

=

mxy·x mxy·y mz·z

X Y Z

= m·

x y z

≡ START First steps Deutsch Rotation

The transformation equation of the planar rotation reads:

X Y

=

cos(ε) −sin(ε) sin(ε) cos(ε)

·

x y

For all spatial transformations there is a spatial rotation, which is described in the following three alternative ways:

(a) with Eulerian angle ε_x or ε_y or ε_z:

X Y Z

=

1  0  0 0 cosεx -sinεx 0 sinεx cosεx

·

x y z

X Y Z

=

cosεy 0 sinεy  0 1  0 -sinεy 0 cosεy

·

x y z

X Y Z

=

cosεz -sinεz  0 sinεz cosεz 0 0  0 1

·

x y z

The total rotation can be viewed as a sequence of three rotations about coordinate axes. Any succession of rotations can be used. Changing the succession of rotations usually changes the result.

(b) with the quadruple of the unit quaternion (q₀ , q₁ , q₂ , q₃). A quaternion usually permits a more elegant description of rotations in three dimensions:

X Y Z

=

q0²+q1²- q2²−q3²    2(q1·q2−q0·q3) 2(q1·q3+q0·q2) 2(q1·q2+q0·q3) q0²−q1²+q2²−q3²    2(q2·q3−q0·q1) 2(q1·q3−q0·q2) 2(q2·q3+q0·q1) q0²−q1²−q2²+q3²

·

x y z

If no unit quaternion with q₀²+q₁²+q₂²+q₃²=1 is given then it is scaled to unity.

(c) with Eulerian axis (e_x , e_y , e_z) as a unit vector and rotation angle ε about it:

X Y Z

=

cosε+ex²(1-cosε) exey(1-cosε)-ezsinε exez(1-cosε)+eysinε exey(1-cosε)+ezsinε    cosε+ey²(1-cosε) eyez(1-cosε)-exsinε exez(1-cosε)-eysinε eyez(1-cosε)+exsinε    cosε+ez²(1-cosε)

·

x y z

The rotation is here described about an oblique axis through the origin of the coordinate system, the so-called Eulerian axis. If no unit vector with e_x²+e_y²+e_z²=1 is given then it is scaled to unit length.

All angles ε, ε_x, ε_y, ε_z are expected between -π = -180° = -200 gon and π = 180° = 200 gon and in the chosen angle unit. They are defined

• for lefthanded systems positive clockwise,
• for righthanded systems positive counterclockwise.

The viewing direction for this is opposite to the rotation axis.

There are alternative conventions in use, mainly in the non-geodetic branches.

Both the parameters q₀, q₁, q₂, q₃and the parameters e_x, e_y, e_z, ε form a compound and must be given in the list of transformation parameters as a consecutive sequence, where the internal succession of the group is arbitrary.

≡ START First steps Deutsch Transvection = shear mapping

The shear mapping is an affine mapping of the plane onto itself, which preserves the area of geometric figures, but angles may change. The transformation equations read:

X Y

=

1 fxy 0  1

·

x y

X Y Z

=

1 fxy 0 0  1 0 0  0 1

·

x y z

X Y Z

=

1 0 fxz 0 1  0 0 0  1

·

x y z

X Y Z

=

1 0 0 fyx 1 0  0 0 1

·

x y z

X Y

=

1 0 fyx 1

·

x y

X Y Z

=

1 0  0 0 1 fyz 0 0  1

·

x y z

X Y Z

=

1 0 0  0 1 0 fzx 0 1

·

x y z

X Y Z

=

1  0 0 0  1 0 0 fzy 1

·

x y z

f_xy=tan(τ_xy)   f_xz=tan(τ_xz)   f_yz=tan(τ_yz)

Either the shear parameters f or the shear angles τ may be specified. All angles are expected between -π/2 = -90° = -100 gon and π/2 = 90° = 100 gon and in the chosen angle unit. Missing shear parameters are treated as zero.

Clue: Two consecutive shear mappings with f=f_yx=-f_xy ≠ 0 are not identical with a rotation with τ=arctan(f). Angles are not preserved:

X Y

=

1   fxy fyx  fyx·fxy+1

·

x y

=

1 -f f 1-f²

·

x y

≡ START First steps Deutsch Planar transformations (2D)

Amongst others, the transformation steps can be combined to the following planar transformations:

Transformation	Parameter	Number
Affine	tx , ty , mx , my , ε , τxy	6
5-Parameter type 1	tx , ty , mx , my , ε	5
5-Parameter type 2	tx , ty , m , τ , ε	5
5-Parameter type 3	tx , ty , ε , mx , my	5
5-Parameter type 4	tx , ty , m , ε , τ	5
Helmert	tx , ty , m , ε	4
with fixed scale	tx , ty , ε	3

≡ START First steps Deutsch Spatial transformations (3D)

Amongst others, the transformation steps can be combined to the following spatial transformations:

Transformation	Parameter	Number
Affine	tx , ty , tz , mx , my , mz, εx , εy , εz , τxy , τxz , τyz	12
9-Parameter Type 1	tx , ty , tz , mx , my , mz , εx , εy , εz	9
9-Parameter Type 2	tx , ty , tz , εx , εy , εz , mx , my , mz	9
Helmert	tx , ty , tz , m , εx , εy , εz	7
with fixed scale	tx , ty , tz , εx , εy , εz	6

As an alternative to the Eulerian angles ε_x , ε_y , ε_z also a quaternion (q₀ , q₁ , q₂ , q₃) or Eulerian axis (e_x , e_y , e_z) and rotation angle ε about it are accepted.

≡ START First steps Deutsch Rotate cuboid about centre axis

The following points with coordinates in a cartesian lefthanded system form the vertices of a cuboid (ABCD=bottom surface, EFGH=top surface):

       X       Y       Z
A   14.034  17.043   8.067
B   23.605  29.759   5.522
C   42.146  16.239   7.807
D   32.585   3.537  10.349
E   14.281  20.222  24.877
F   23.842  32.924  22.335
G   42.393  19.418  24.617
H   32.841   6.711  27.167

It must be rotated by 45° about its centre axis parallel to AE (⇑ figure) counterclockwise, as seen from above. The point to be held fixed during rotation may be the centre point M of the cuboid

M   28.2159 18.2316 16.3426

computed as the mean of all coordinates e.g. with , or the centre point of the bottom or top surface. The Eulerian axis e may be described by the vector

AE=	14.281-14.034 20.222-17.043 24.877-8.067

Since for lefthanded systems, when looking opposite to the rotation axial direction, the rotation angle is positive clockwise (⇑ Rotation ), the rotation angle ε=−45° must be specified. As an alternative, you could use EA instead of AE as an axis vector. The you should use ε = 45° .

and Compute

The result is

X Y Z

=

-2.33842866 23.6949266 -4.44667340

+

0.70716782 0.69550488 -0.12722668 -0.69393365 0.71721800 0.06367434 0.13553508 0.04325843 0.98982774

.

x y z

PName	X	Y	Z
A	18.4131166747	26.6934690300	6.1776196987
B	34.3492519040	29.0099229729	5.5057885573
C	37.7669114811	6.5924075628	9.6956467711
D	21.8479666784	4.2788643123	10.3664693981
E	18.6601166747	29.8724690300	22.9876196987
F	34.5790614774	32.1860122804	22.3167970717
G	38.0139114811	9.7714075628	26.5056467711
H	22.0968358509	7.4485422141	27.1853915435

Perform a using these coordinate lists. All residuary misclosures must be zero, independent of the weighting.

Did you know? You must not sign in to IN DUBIO PRO GEO, therefore you work anonymously.

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