atan2

Four-quadrant inverse tangent of fixed-point values

Syntax

z = atan2(y,x)

Description

z = atan2(y,x) returns the four-quadrant arctangent of fi inputs y and x.

Examples

Calculate Arctangent of Fixed-Point Input Values

Use the atan2 function to calculate the arctangent of unsigned and signed fixed-point input values.

Unsigned Input Values

This example uses unsigned, 16-bit word length values.

y = fi(0.125,0,16);
x = fi(0.5,0,16);  
z = atan2(y,x)

z = 
    0.2450

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Unsigned
            WordLength: 16
        FractionLength: 15

Signed Input Values

This example uses signed, 16-bit word length values.

y = fi(-0.1,1,16);
x = fi(-0.9,1,16);  
z = atan2(y,x)

z = 
   -3.0309

          DataTypeMode: Fixed-point: binary point scaling
            Signedness: Signed
            WordLength: 16
        FractionLength: 13

Input Arguments

y — y-coordinates

scalar | vector | matrix | multidimensional array

y-coordinates, specified as a scalar, vector, matrix, or multidimensional array.

y and x can be real-valued, signed or unsigned scalars, vectors, matrices, or N-dimensional arrays containing fixed-point angle values in radians. The inputs y and x must be the same size. If they are not the same size, at least one input must be a scalar value. Valid data types of y and x are:

• fi single

• fi double

• fi fixed-point with binary point scaling

• fi scaled double with binary point scaling

Data Types: fi

x — x-coordinates

scalar | vector | matrix | multidimensional array

x-coordinates, specified as a scalar, vector, matrix, or multidimensional array.

y and x can be real-valued, signed or unsigned scalars, vectors, matrices, or N-dimensional arrays containing fixed-point angle values in radians. The inputs y and x must be the same size. If they are not the same size, at least one input must be a scalar value. Valid data types of y and x are:

• fi single

• fi double

• fi fixed-point with binary point scaling

• fi scaled double with binary point scaling

Data Types: fi

Output Arguments

z — Four-quadrant arctangent

scalar | vector | matrix | multidimensional array

Four-quadrant arctangent, returned as a scalar, vector, matrix, or multidimensional array.

z is the four-quadrant arctangent of y and x. The numerictype of z depends on the signedness of y and x:

• If either y or x is signed, then z is a signed, fixed-point number in the range [–pi,pi]. It has a 16-bit word length and 13-bit fraction length (numerictype(1,16,13)).

• If both y and x are unsigned, then z is an unsigned, fixed-point number in the range [0,pi/2]. It has a 16-bit word length and 15-bit fraction length (numerictype(0,16,15)).

The output, z, is always associated with the default fimath.

More About

Four-Quadrant Arctangent

The four-quadrant arctangent is defined as follows, with respect to the atan function:

$$\begin{array}{l}\text{atan2}(y,x)=\{\begin{array}{l}\text{atan}\left(\frac{y}{x}\right)x>0\\ \pi +\text{atan}\left(\frac{y}{x}\right)y\ge 0,x<0\\ -\pi +\text{atan}\left(\frac{y}{x}\right)y<0,x<0\\ \frac{\pi }{2}y>0,x=0\\ -\frac{\pi }{2}y<0,x=0\\ 0y=0,x=0\end{array}\\ \end{array}$$

Algorithms

The atan2 function computes the four-quadrant arctangent of fixed-point inputs using an 8-bit lookup table as follows:

1. Divide the input absolute values to get an unsigned, fractional, fixed-point, 16-bit ratio between 0 and 1. The absolute values of y and x determine which value is the divisor.

   The signs of the y and x inputs determine in what quadrant their ratio lies. The input with the larger absolute value is used as the denominator, thus producing a value between 0 and 1.

   

2. Compute the table index, based on the 16-bit, unsigned, stored integer value:

   1. Use the 8 most-significant bits to obtain the first value from the table.

   2. Use the next-greater table value as the second value.

3. Use the 8 least-significant bits to interpolate between the first and second values using nearest neighbor linear interpolation. This interpolation produces a value in the range [0, pi/4).

4. Perform octant correction on the resulting angle, based on the values of the original y and x inputs.

This arctangent calculation is accurate only to within the top 16 most-significant bits of the input.

fimath Propagation Rules

The atan2 function ignores and discards any fimath attached to the inputs. The output, z, is always associated with the default fimath.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.