Generating angular coordinate of raster scan of polar-coordinate addressed memory

In combination:

• a memory for storing data addressed in first and second coordinates, which second coordinates are angular coordinates;
• raster generator means for generating a description of raster scan in x and y Cartesian coordinates;
• scan conversion means for converting said description of raster scan in x and y Cartesian coordinates to a description of raster scan in said first and second coordinates, which description in said first and second coordinates is used to address said memory for storing data, thereby to raster scan the data stored in said memory;
• means, included in said scan conversion means, for supplying from read-only memory respective logarithmic responses to said x and y Cartesian coordinates applied as input addresses;
• means, included in said scan conversion means, for linearly combining those logarithmic responses to obtain logarithms descriptive of one of said x and y Cartesian coordinates divided by the other;
• means included in said scan conversion means, for supplying from further read-only memory said angular coordinates as the arc tangents of the anti-logarithms of said logarithms descriptive of one of said x and y Cartesian coordinates divided by the other, which logarithms are applied as input addresses to said further read-only memory;
• and
• means, included in said scan conversion means, for determining said second coordinates from said arc tangents.

In combination:

• a memory for storing data addressed in angular coordinates and in coordinates orthogonal to said angular coordinates;
• raster generator means for generating a description of raster scan in a two's complement x coordinate and a two's complement y coordinate;
• means responsive to said x and y coordinates for generating said coordinates orthogonal to said angular coordinates, which are applied to said memory for storing data for addressing that memory in coordinates orthogonal to said angular coordinates;
• means for exclusive-ORing the most significant or sign bit of said x coordinate with each of its other bits, and
• means for obtaining logarithms of the signal samples thus generated;
• means for exlusive-ORing the most significant or sign bit of said y coordinate with each of its other bits, and
• means for obtaining logarithms of the reciprocal of the signal samples thus generated;
• means for linearly combining the logarithms, and
• means for obtaining from table look-up memory the arc tangents of the anti-logarithms of the signal samples thus generated were x and y descriptive of one quadrant of the angular coordinates;
• means for specifying said angular coordinates in two's complement form, including
• means responsive to the most significant bit of one of said x and y coordinates for determining the most significant bit of said angular coordinates absent rotation of the data read from said memory storing data, including
• means responsive to exclusive OR of the most significant bits of said x and y coordinates for determining the secondmost significant bit of said angular coordinates absent rotation of the data read from said memory for storing data, including
• means dependent on that secondmost significant bit for selectively complementing said arc tangents of anti-logarithms of signal samples to provide the bits of lesser significance in said angular coordinates absent rotation of the data read from said memory for storing data, and where the data read from said memory for storing data are to be rotated including
• means for offsetting the two's complement number thusfar formed by combination with another two's complement number indicative of the degree of data rotation, thereby to generate addresses in angular coordinates which are applied to said memory for storing data for completing the addressing of that memory.

In the combination of

• means for generating a description of a scanning point in Cartesian coordinates, one of said Cartesian coordinates being an x coordinate in two's complement form and the other being a y coordinate in two's complement form, and means for converting said Cartesian coordinates to angular coordinates, the improvement wherein said means for converting comprises:
• means for calculating said angular coordinates from the logarithm of the magnitude of one of said x and y coordinates divided by the magnitude of the other;
• means linearly combining logarithmic functions of said x and y coordinates for approximating the logarithm of one of said x and y coordinates divided by the other;
• means for providing table look-up of logarithmic functions of said x and y coordinates responsive to first and second signals respectively, which means for providing avoids the problem of dividing by zero by including, in addition to
• read-only memory means for storing logarithm look-up tables,
• means exclusive-ORing the most significant bit of said x coordinate with each of its less significant bits to generate a first set of input signals for said read-only memory means, and
• means exclusive-ORing the most significant bit of said y coordinate with each of its less significant bits to generate a second set of input signals for said read-only memory means.

Apparatus for calculating a function of the angle between first and second orthogonal vectors, comprising:

• means for generating a digital electric signal representation of the amplitude of said first vector;
• means for generating a digital electric signal representation of the amplitude said second vector;
• read-only memory receptive of said digital electric signal representation of the amplitude said first vector for responding with a digital electric signal representation of the logarithm of the amplitude of said first vector;
• read-only-memory receptive of said digital electric signal representation of the amplitude of said second vector for responding with a digital electric signal representation of the logarithm of the reciprocal of the amplitude of said second vector;
• an adder adding the digital electronic signal representations of the logarithm of the amplitude of the first vector and of the reciprocal of the amplitude of the second vecotor for developing a digital electric signal representation of the quotient of the amplitudes of said first and second vectors;
• and
• read-only memory receptive of the digital electric signal representation of the quotient of the amplitudes of said first and second vectors for supplying a digital electric signal representation of the associated value of the function of the angle between said first and second vectors.