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Building a Local Coordinate Transformation Matrix in 3D Using MATLAB

by alpsdev




When working with 3D structural elements such as beams, frames, or trusses, it is often necessary to define a local coordinate system aligned with the element. This local frame allows for easier formulation of element stiffness, loads, and deformations.

In this blog, we'll walk through how to compute a local transformation matrix from two given points (start and end nodes) using MATLAB, and ensure it is a right-handed orthonormal system.


Problem Statement

Given two nodes:

  • Node i: (x1, y1, z1)

  • Node j: (x2, y2, z2)


We want to build a transformation matrix T that defines a local coordinate system aligned with the line connecting these two nodes.

  • Local x'-axis: points from node i to node j.

  • Local y'-axis: constructed to be perpendicular to x'-axis.

  • Local z'-axis: completes the right-handed system.

The resulting matrix T will have:

  • Each row representing a unit basis vector (x', y', z') in the global coordinate system.


Step-by-Step MATLAB Script

% Node coordinates
node_i_coord = [0, 0, 0];
node_j_coord = [5, 3, 2];

% Compute local x' axis (from node i to j)
vec_x = node_j_coord - node_i_coord;
dx = vec_x / norm(vec_x); % Normalize to unit vector

% Define global reference axes
global_Z = [0, 0, 1];
global_Y = [0, 1, 0];

% Check for near vertical case
if abs(dot(dx, global_Z)) > 0.99
    % x' is almost vertical; use Y-axis to construct local z'
    dz = cross(dx, global_Y);
else
    % Normal case; use Z-axis to construct local z'
    dz = cross(dx, global_Z);
end

% Normalize local z' axis
dz = dz / norm(dz);

% Compute local y' axis by cross product
% (Ensure right-handed system: dy = dz x dx)
dy = cross(dz, dx);
dy = dy / norm(dy);

% Assemble Transformation matrix
T = [dx; dy; dz];

Concept Behind Each Step

  1. Local x'-axis (dx):

    • Defined by the direction from node i to node j.

    • Normalized to ensure unit length.

  2. Choosing Reference Axis:

    • To define the other two axes (y', z'), we need a second reference direction.

    • Normally, we use the global Z-axis ([0 0 1]).

    • However, if dx is almost vertical, the cross product with Z becomes unstable.

    • In that case, we switch to Y-axis ([0 1 0]).

  3. Local z'-axis (dz):

    • Cross dx with the reference axis.

    • Normalize to ensure unit length.

  4. Local y'-axis (dy):

    • Cross dz and dx to form the third axis.

    • This ensures a right-handed orthonormal system.

  5. Building T Matrix:

    • Stack dx, dy, dz as rows.

    • Now T transforms vectors from local to global coordinates.

Why This Method is Robust

  • Handles near vertical elements: switching reference axes automatically.

  • Ensures orthogonality: by using cross products.

  • Normalized basis vectors: each axis has unit length.

You can check correctness by confirming that:

T * T' % Should be approximately Identity matrix

Final Thoughts

This method is essential for constructing the transformation matrices used in:

  • Finite Element Analysis (FEA)

  • Frame element stiffness matrix assembly

  • Post-processing stresses and strains in local coordinates

Building a reliable transformation matrix ensures your element behavior is correctly represented no matter how they are oriented in space.


Bonus





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