Coordinate systems

Introduction

In the three dimensions of our world you can define different coordinate systems. In our auditory labs a subject is placed with his head in the center of the speaker setup (a sphere or semi-circle). We define all coordinate systems with respect to the default position of the head of the subject. Normally we are interested in the direction of a stimulus and we therefore define the directions with the angles azimuth and elevation in a double polar coordinate system. This system is a non-conventional coordinate system that is not used often beyond auditory experiments. The standard system is the Spherical Coordinate System. Both the Azimuth and Elevation are calculated differently in these two systems.

The head tracking system uses three orthogonal axis: Horizontal, Frontal, Vertical. Sometimes we refer to them as X, Y and Z. In the following paragraphs each of the double polar and Cartesian coordinate systems are defined as they are used in our auditory labs.

When your are interested in the Spherical coordinate system you have to ask Wikipedia or other sources.

Convention for double polar coordinates

The double polar coordinates are called Azimuth and Elevation. The angle Azimuth defines a semi-circle parallel to the YZ plane. The angle Elevation defines a circle parallel to the XY plane. The intersection of the Azimuth semi-circle and the full Elevation circle is the target point.

azimuth

Azimuth is the angle (theta) with the YZ plane for +Y and pi - theta for -Y.
sign(Azimuth) is sign(X)

elevation

Elevation is the angle with the XY plane.
sign(Elevation) is sign(Z)

Transformation of double polar to Cartesian coordinates

X = R * sin(azimuth)
Z = R * sin(elevation)
Y = sign(cos(azimuth)*cos(elevation)) * sqrt(R^2 - X^2 - Z^2)

Transformation of Cartesian to double polar coordinates

azimuth   = atan2 (X, sqrt (Z^2 + Y^2))
elevation = atan2 (Z, sqrt (X^2 + Y^2))

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

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

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

Conversion function in Matlab

We have defined two conversion functions between double polar and Cartesian coordinates.

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

[X,Y,Z] = Function double_polar2cart(azimuth, elevation, r)

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