Consistent System of Rectangular
and Spherical Coordinates for
Electrocardiology and Magnetocardiology
Relative to the natural observation planes of the patient (i.e., the frontal plane observed from the front, the sagittal plane observed from left, and the transverse plane observed from above), only the sagittal plane is observed from the positive side.
The spherical coordinate system, fixed to this coordinate system with the generally accepted axes, results in an unfamiliar orientation.
The rectangular coordinate system should be right-handed to be consistent with conventions in the physical sciences and to permit straightforward application of the standard equations used in vector analysis and electromagnetism.
The three coordinate planes are the xy, yz, and zx planes.
Each plane is viewed from positive side.
Angles in the xy, yz, and zx planes are measured in the positive direction (counterclockwise) from the x, y, and z axes, respectively, with a range of either 0° to 360° or 0° to ±180° , with negative angles being measured in a clockwise direction from the axis.
The four quadrants in each coordinate plane are specified in a positive, counterclockwise, sequence:
The radius vector is described by the symbol r. The angles θ and φ are called colatitude and longitude, respectively. The angle θ is also called polar angle because it is an angle dimensioned from the pole (i.e. z axis). These requirements for rectangular and spherical polar coordinates are based on existing mathematical conventions. This mathematically consistent coordinate system is illustrated in Figure A.3A.
In the illustrative spherical coordinate system, the vector magnitude is represented by the symbol M (being the same as the radius vector r in the spherical polar coordinate system). The angles E and A are called the elevation and azimuth, respectively. This coordinate system is illustrated in Figure A.3B. (Note, that when the angles elevation and azimuth are those used in connection with the AHA coordinate system, they are represented by symbols V and H and they differ from those of the consistent system introduced in this chapter.)
The angles elevation and azimuth correspond exactly to the latitude and longitude angles, respectively, used in geography. Therefore, ordinary (and familiar) geographic map projection techniques can be immediately applied to maps of electric potential and magnetic field over the entire torso surface, as described in Figure A.4.
In the rectangular coordinates, the x and z coordinates in the consistent system have opposite polarity to those in the AHA system. However, the consistent system and the AHA system have identical vector loop displays.
In the spherical coordinates, the elevation angles (E and V) are the same in both systems except for different polarity. The azimuth angles (A and H) have the same polarity in both systems, but because of the different reference axis the values in the consistent system differ by 90° from the values in the AHA system. The vector magnitude M is, of course, the same in both systems. (Note, that in the existing literature one may find other definitions for the elevation and azimuth angle than those of the AHA.)
A.6 RECTANGULAR ABC COORDINATES
In addition to the XYZ coordinate system in magnetocardiography, another right-handed coordinate system is needed that is oriented more symmetrically in relation to the frontal plane (Malmivuo, 1976). The three axes of this coordinate system are selected to be the three edges of a cube whose diagonal is parallel to the x axis. This system is called the ABC coordinate system. Figure A.5 shows the orientation of these axes in relation to the x, y, and z axes. The ABC axes form an angle of 35° 15' 52" with the yz plane, and the angle between their projections on this plane is 120°. The projection of the A axis is parallel to the z axis.
The components of a vector in the ABC coordinate system may be transformed to the XYZ coordinate system with the following linear transformation (Equation A.2):
The components of a vector in the XYZ coordinate system may be transformed to the ABC coordinate system with the following linear transformation (Equation A.3):
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