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3-D rigid geometric transformation

Since R2022b


    A rigidtform3d object stores information about a 3-D rigid geometric transformation and enables forward and inverse transformations.


    You can create a rigidtform3d object in these ways:

    • imregtform — Estimates a geometric transformation that maps a moving image to a fixed image using similarity optimization.

    • Functions in other toolboxes that return geometric transformations, including but not limited to:

    • The rigidtform3d function described here.


    tform = rigidtform3d creates a rigidtform3d object that performs an identity transformation.

    tform = rigidtform3d(R,Translation) creates a rigidtform3d object that performs a rigid transformation based on the specified values of the R and Translation properties. These properties indicate the rotation matrix and the amounts of translation in the x-, y-, and z-directions.


    tform = rigidtform3d(eulerAngles,Translation) creates a rigidtform3d object that performs a rigid transformation based on Euler angles and the specified value of the Translation property.

    tform = rigidtform3d(A) creates a rigidtform3d object and sets the property A as the specified 3-D rigid transformation matrix.

    tform = rigidtform3d(tformIn) creates a rigidtform3d object from another geometric transformation object, tformIn, that represents a valid 3-D rigid geometric transformation.

    Input Arguments

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    Euler angles in x,y,z-order in degrees, specified as a 3-element numeric vector of the form [rx ry rz]. The Euler angles set the R property as a product of three rotation matrices according to:

     Rx = [1 0 0; 0 cosd(rx) -sind(rx); 0 sind(rx) cosd(rx)];
     Ry = [cosd(ry) 0 sind(ry); 0 1 0; -sind(ry) 0 cosd(ry)];
     Rz = [cosd(rz) -sind(rz) 0; sind(rz) cosd(rz) 0; 0 0 1];
      R = Rz*Ry*Rx;

    Data Types: double | single

    Rigid 3-D geometric transformation, specified as an affinetform3d object, rigidtform3d object, simtform3d object, or transltform3d object.


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    Forward 3-D rigid transformation, specified as a 4-by-4 numeric matrix. When you create the object, you can also specify A as a 3-by-4 numeric matrix. In this case, the object concatenates the row vector [0 0 0 1] to the end of the matrix, forming a 4-by-4 matrix. The default of A is the identity matrix.

    The matrix A transforms the point (u, v, w) in the input coordinate space to the point (x, y, z) in the output coordinate space using the convention:


    For a rigid transformation, A has the form:


    where each element R(i,j) is element (i, j) of the rotation matrix specified by the R property. tx, ty, and tz are the amount of translation in the x-, y-, and z-directions, respectively, and correspond to the Translation property.

    Data Types: double | single

    Rotation matrix, specified as a 3-by-3 numeric matrix. The rotation matrix has the effect of rotating about the z-axis first, then the y-axis, and then the x-axis.

    Amount of translation, specified as a 3-element numeric vector of the form [tx ty tz].

    Data Types: double | single

    This property is read-only.

    Dimensionality of the geometric transformation for both input and output points, specified as 3.

    Data Types: double

    Object Functions

    invertInvert geometric transformation
    outputLimitsFind output spatial limits given input spatial limits
    transformPointsForwardApply forward geometric transformation
    transformPointsInverseApply inverse geometric transformation


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    Specify Euler angles and amounts of translation.

    angles = [30 0 90];
    translation = [10 20.5 15];

    Create a rigidtform3d object that performs the specified rotation and translation.

    tform = rigidtform3d(angles,translation)
    tform = 
      rigidtform3d with properties:
        Dimensionality: 3
           Translation: [10 20.5000 15]
                     R: [3x3 double]
                     A: [     0   -0.8660    0.5000   10.0000
                         1.0000         0         0   20.5000
                              0    0.5000    0.8660   15.0000
                              0         0         0    1.0000]

    Examine the value of the A property.

    ans = 4×4
             0   -0.8660    0.5000   10.0000
        1.0000         0         0   20.5000
             0    0.5000    0.8660   15.0000
             0         0         0    1.0000

    Extended Capabilities

    Version History

    Introduced in R2022b

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