Coordinate systems

Convention for Cartesian coordinates

• Middle of the sphere is (0, 0, 0)
• X is left (-) to right (+)
• Y is back (-) to front (+)
• Z is bottom (-) to top (+)

N.B. The Z-hat vector is the outer product of the X-hat vector and Y-hat vector (right hand rule).

The convention for head tracking coordinates is equal to Cartesian coordinates

• Horizontal = X
• Frontal = Y
• Vertical = Z

Convention for spherical coordinates

In the spherical coordinate system the position of a given point in space is specified by three numbers, (r, θ, φ): the radial distance of the radial line r connecting the point to the fixed point of origin (which is located on a fixed polar axis, or zenith direction axis, or z-axis); the polar angle θ of the radial line r; and the azimuthal angle φ of the radial line r.

• The elevation or polar angle θ is the angle with Z-hat.
• The elevation runs from 0 to pi.
• The azimuth or azimuthal angle φ is the angle of the projection to the XY plane with X-hat in the anti clockwise direction.
• The azimuth runs from 0 to 2pi.

This is the so-called physics convention (ISO 80000-2:2019).

Convention for double polar coordinates

• Azimuth is the angle of the projection to the XY plane with Y-hat.
• Elevation is angle of the projection to the YZ plane with Y-hat.

• Azimuth is between -pi and pi and positive in the X-hat direction.
• Elevation is between -pi/2 and pi/2 and positive in the Z-hat direction.
• In the negative Y direction the Elevation is mirrored (Elevation = angle with minus Y-hat).

Conversion from Cartesian to Double Polar

% Double polar coördinates is a non-standard coördinate system used for audiological purposes only.

The procedure for transformation is the following:

• 1 Project the Cartesian coordinates to the XY-plane.
• 2 This forms a two dimensional vector in the XY-plane.
• 3 Take the angle of this vector with the Y-hat. This is the azimuth.
• 4 Project the Cartesian coordinates to the YZ-plane.
• 5 Take the angle of this vector with the Y-hat. This is the elevation.
• 6 Angles larger than pi/2 are folded back by mirroring

How to get the angle between two vectors

The algorithm makes use of: dotproduct(V1,V2) = |V1|*|V2|*cos(angle) to determine the angle.

function theta = getAngle(V1, V2)
    dotProduct = dot(V1, V2);   
    norm1 = norm(V1);
    norm2 = norm(V2);    
    if norm1 * norm2 > 0
        cosTheta = dotProduct / (norm1 * norm2);
        theta = acos(cosTheta);        
    else
        theta = 0;
    end    
end

cart2double_polar(X, Y, Z)

function [azimuth, elevation, r] = cart2double_polar(X, Y, Z)

    r = sqrt(X^2+Y^2+Z^2);
            
    %azimuth: projection to XY plane, angle with Y_hat
    az_angleWithY_hat = getAngle([X,Y,0],[0,1,0]);

    %elevation: projection to YZ plane, angle with Y_hat
    el_angleWithY_hat = getAngle([0,Y,Z],[0,1,0]);
    
    % take the quadrants into account
    switch findQuadrant(X, Y) % Quadrants are numbered anti-clockwise
        case {1,4}
            azimuth = az_angleWithY_hat;            
        case {2, 3}
            azimuth = - az_angleWithY_hat;
        case 0 % (X=0 and Y=0)
            azimuth = 0;
    end

    switch findQuadrant(Y, Z) 
        case 1 
            elevation = el_angleWithY_hat;
        case 2
            elevation = pi - el_angleWithY_hat;
        case 3
            elevation = el_angleWithY_hat - pi;
        case 4
            elevation = - el_angleWithY_hat;
        case 0 % (Y=0 and Z=0)
            elevation = 0;
    end