Class TrigTools

java.lang.Object
com.github.tommyettinger.digital.TrigTools

public final class TrigTools extends Object
Various trigonometric approximations, using a lookup table for sin(), cos(), and tan(), and Taylor series for their inverses. This supplies variants for radians, degrees, and turns. This also has an atan2() approximation defined with output in radians, degrees, and turns.
  • Field Summary

    Fields
    Modifier and Type
    Field
    Description
    static final float
    Multiply by this to convert from degrees to radians.
    static final float
    PI divided by 2f.
    static final float
    The float value that is closer than any other to pi, the ratio of the circumference of a circle to its diameter.
    static final float
    1.0f divided by PI.
    static final float
    2f times PI; the same as TAU.
    static final float
    PI divided by 4f.
    static final float
    Multiply by this to convert from radians to degrees.
    static final float[]
    A precalculated table of 16384 floats, corresponding to the y-value of points on the unit circle, ordered by increasing angle.
    static final int
    The bitmask that can be used to confine any int to wrap within TABLE_SIZE.
    static final int
    The size of SIN_TABLE, available separately from the table's length for convenience.
    static final float
    2f times PI; the same as PI2.
  • Method Summary

    Modifier and Type
    Method
    Description
    static float
    acos(float a)
    Returns arccosine in radians; less accurate than Math.acos but may be faster.
    static float
    acosDeg(float a)
    Returns arccosine in degrees.
    static float
    acosTurns(float a)
    Returns arccosine in turns.
    static float
    asin(float a)
    Returns arcsine in radians; less accurate than Math.asin but may be faster.
    static float
    asinDeg(float a)
    Returns arcsine in degrees.
    static float
    asinTurns(float a)
    Returns arcsine in turns.
    static float
    atan(float i)
    Arc tangent approximation with very low error, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise).
    static float
    atan2(float y, float x)
    Close approximation of the frequently-used trigonometric method atan2, using radians.
    static float
    atan2Deg(float y, float x)
    Close approximation of the frequently-used trigonometric method atan2, using positive or negative degrees.
    static float
    atan2Deg360(float y, float x)
    Close approximation of the frequently-used trigonometric method atan2, using non-negative degrees only.
    static float
    atan2Turns(float y, float x)
    Close approximation of the frequently-used trigonometric method atan2, using non-negative turns only.
    static float
    atanDeg(float i)
    Arc tangent approximation returning a value measured in positive or negative degrees, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise).
    static float
    atanTurns(float i)
    Arc tangent approximation with very low error, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise).
    static double
    atanUnchecked(double i)
    A variant on atan(float) that does not tolerate infinite inputs for speed reasons.
    static double
    atanUncheckedDeg(double i)
    A variant on atanDeg(float) that does not tolerate infinite inputs for speed reasons.
    static double
    A variant on atanTurns(float) that does not tolerate infinite inputs for speed reasons.
    static float
    cos(float radians)
    Returns the cosine in radians from a lookup table.
    static float
    cosDeg(float degrees)
    Returns the cosine in degrees from a lookup table.
    static float
    cosTurns(float turns)
    Returns the cosine in turns from a lookup table.
    static float
    sin(float radians)
    Returns the sine in radians from a lookup table.
    static float
    sinDeg(float degrees)
    Returns the sine in degrees from a lookup table.
    static float
    sinTurns(float turns)
    Returns the sine in turns from a lookup table.
    static float
    tan(float radians)
    Returns the tangent in radians from a lookup table.
    static float
    tanDeg(float degrees)
    Returns the tangent in degrees from a lookup table.
    static float
    tanTurns(float turns)
    Returns the tangent in turns from a lookup table.

    Methods inherited from class java.lang.Object

    clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
  • Field Details

    • PI

      public static final float PI
      The float value that is closer than any other to pi, the ratio of the circumference of a circle to its diameter.
      See Also:
    • PI_INVERSE

      public static final float PI_INVERSE
      1.0f divided by PI.
      See Also:
    • PI2

      public static final float PI2
      2f times PI; the same as TAU.
      See Also:
    • TAU

      public static final float TAU
      2f times PI; the same as PI2.
      See Also:
    • HALF_PI

      public static final float HALF_PI
      PI divided by 2f.
      See Also:
    • QUARTER_PI

      public static final float QUARTER_PI
      PI divided by 4f.
      See Also:
    • TABLE_SIZE

      public static final int TABLE_SIZE
      The size of SIN_TABLE, available separately from the table's length for convenience.
      See Also:
    • TABLE_MASK

      public static final int TABLE_MASK
      The bitmask that can be used to confine any int to wrap within TABLE_SIZE. Any accesses to SIN_TABLE with an index that could be out of bounds should probably be wrapped using this, as with SIN_TABLE[index & TABLE_MASK].
      See Also:
    • radiansToDegrees

      public static final float radiansToDegrees
      Multiply by this to convert from radians to degrees.
      See Also:
    • degreesToRadians

      public static final float degreesToRadians
      Multiply by this to convert from degrees to radians.
      See Also:
    • SIN_TABLE

      public static final float[] SIN_TABLE
      A precalculated table of 16384 floats, corresponding to the y-value of points on the unit circle, ordered by increasing angle. This should not be mutated, but it can be accessed directly for things like getting random unit vectors, or implementing the "sincos" method (which assigns sin() to one item and cos() to another).
      A quick way to get a random unit vector is to get a random 14-bit number, as with int angle = random.nextInt() >>> 18;, look up angle in this table to get y, then look up (angle + 4096) & 16383 to get x.
  • Method Details

    • sin

      public static float sin(float radians)
      Returns the sine in radians from a lookup table. For optimal precision, use radians between -PI2 and PI2 (both inclusive).
      Parameters:
      radians - an angle in radians, where 0 to PI2 is one rotation
    • cos

      public static float cos(float radians)
      Returns the cosine in radians from a lookup table. For optimal precision, use radians between -PI2 and PI2 (both inclusive).
      Parameters:
      radians - an angle in radians, where 0 to PI2 is one rotation
    • tan

      public static float tan(float radians)
      Returns the tangent in radians from a lookup table. For optimal precision, use radians between -PI2 and PI2 (both inclusive).
      Parameters:
      radians - an angle in radians, where 0 to PI2 is one rotation
    • sinDeg

      public static float sinDeg(float degrees)
      Returns the sine in degrees from a lookup table. For optimal precision, use degrees between -360 and 360 (both inclusive).
      Parameters:
      degrees - an angle in degrees, where 0 to 360 is one rotation
    • cosDeg

      public static float cosDeg(float degrees)
      Returns the cosine in degrees from a lookup table. For optimal precision, use degrees between -360 and 360 (both inclusive).
      Parameters:
      degrees - an angle in degrees, where 0 to 360 is one rotation
    • tanDeg

      public static float tanDeg(float degrees)
      Returns the tangent in degrees from a lookup table. For optimal precision, use degrees between -360 and 360 (both inclusive).
      Parameters:
      degrees - an angle in degrees, where 0 to 360 is one rotation
    • sinTurns

      public static float sinTurns(float turns)
      Returns the sine in turns from a lookup table. For optimal precision, use turns between -1 and 1 (both inclusive).
      Parameters:
      turns - an angle in turns, where 0 to 1 is one rotation
    • cosTurns

      public static float cosTurns(float turns)
      Returns the cosine in turns from a lookup table. For optimal precision, use turns between -1 and 1 (both inclusive).
      Parameters:
      turns - an angle in turns, where 0 to 1 is one rotation
    • tanTurns

      public static float tanTurns(float turns)
      Returns the tangent in turns from a lookup table. For optimal precision, use turns between -1 and 1 (both inclusive).
      Parameters:
      turns - an angle in turns, where 0 to 1 is one rotation
    • atanUnchecked

      public static double atanUnchecked(double i)
      A variant on atan(float) that does not tolerate infinite inputs for speed reasons. This can be given a double parameter, but is otherwise the same as atan(float), and returns a float like that method. It uses the same approximation, from sheet 11 of "Approximations for Digital Computers." This is mostly meant to be used inside atan2(float, float), but it may be a tiny bit faster than atan(float) in other code.
      Parameters:
      i - any finite double or float, but more commonly a float
      Returns:
      an output from the inverse tangent function in radians, from -HALF_PI to HALF_PI inclusive
    • atanUncheckedTurns

      public static double atanUncheckedTurns(double i)
      A variant on atanTurns(float) that does not tolerate infinite inputs for speed reasons. This can be given a double parameter, but is otherwise the same as atanTurns(float), but returns a double in case external code needs higher precision. It uses the same approximation, from sheet 11 of "Approximations for Digital Computers." This is mostly meant to be used inside atan2Turns(float, float), but it may be a tiny bit faster than atanTurns(float) in other code.
      Parameters:
      i - any finite double or float, but more commonly a float
      Returns:
      an output from the inverse tangent function in turns, from -0.5 to 0.5 inclusive
    • atanUncheckedDeg

      public static double atanUncheckedDeg(double i)
      A variant on atanDeg(float) that does not tolerate infinite inputs for speed reasons. This can be given a double parameter, but is otherwise the same as atanDeg(float), and returns a float like that method. It uses the same approximation, from sheet 11 of "Approximations for Digital Computers." This is mostly meant to be used inside atan2(float, float), but it may be a tiny bit faster than atanDeg(float) in other code.
      Parameters:
      i - any finite double or float, but more commonly a float
      Returns:
      an output from the inverse tangent function in degrees, from -90 to 90 inclusive
    • atan2

      public static float atan2(float y, float x)
      Close approximation of the frequently-used trigonometric method atan2, using radians. Average error is 1.057E-6 radians; maximum error is 1.922E-6. Takes y and x (in that unusual order) as floats, and returns the angle from the origin to that point in radians. It is about 4 times faster than Math.atan2(double, double) (roughly 15 ns instead of roughly 60 ns for Math, on Java 8 HotSpot).
      Credit for this goes to the 1955 research study "Approximations for Digital Computers," by RAND Corporation. This is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise. The algorithms on sheets 8-10 are faster, but only by a very small degree, and are considerably less precise. That study provides an atan(float) method, and that cleanly translates to atan2().
      Parameters:
      y - y-component of the point to find the angle towards; note the parameter order is unusual by convention
      x - x-component of the point to find the angle towards; note the parameter order is unusual by convention
      Returns:
      the angle to the given point, in radians as a float; ranges from -PI to PI
    • atan2Deg

      public static float atan2Deg(float y, float x)
      Close approximation of the frequently-used trigonometric method atan2, using positive or negative degrees. Average error is ??? degrees; maximum error is ???. Takes y and x (in that unusual order) as floats, and returns the angle from the origin to that point in degrees.
      Credit for this goes to the 1955 research study "Approximations for Digital Computers," by RAND Corporation. This is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise. The algorithms on sheets 8-10 are faster, but only by a very small degree, and are considerably less precise. That study provides an atan(float) method, and that cleanly translates to atan2Deg().
      Parameters:
      y - y-component of the point to find the angle towards; note the parameter order is unusual by convention
      x - x-component of the point to find the angle towards; note the parameter order is unusual by convention
      Returns:
      the angle to the given point, in degrees as a float; ranges from -180 to 180
    • atan2Deg360

      public static float atan2Deg360(float y, float x)
      Close approximation of the frequently-used trigonometric method atan2, using non-negative degrees only. Average error is ??? degrees; maximum error is ???. Takes y and x (in that unusual order) as floats, and returns the angle from the origin to that point in degrees.
      Credit for this goes to the 1955 research study "Approximations for Digital Computers," by RAND Corporation. This is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise. The algorithms on sheets 8-10 are faster, but only by a very small degree, and are considerably less precise. That study provides an atan(float) method, and that cleanly translates to atan2Deg360().
      Parameters:
      y - y-component of the point to find the angle towards; note the parameter order is unusual by convention
      x - x-component of the point to find the angle towards; note the parameter order is unusual by convention
      Returns:
      the angle to the given point, in degrees as a float; ranges from 0 to 360
    • atan2Turns

      public static float atan2Turns(float y, float x)
      Close approximation of the frequently-used trigonometric method atan2, using non-negative turns only. Average error is ??? degrees; maximum error is ???. Takes y and x (in that unusual order) as floats, and returns the angle from the origin to that point in turns.
      Credit for this goes to the 1955 research study "Approximations for Digital Computers," by RAND Corporation. This is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise. The algorithms on sheets 8-10 are faster, but only by a very small degree, and are considerably less precise. That study provides an atan(float) method, and that cleanly translates to atan2Turns().
      Parameters:
      y - y-component of the point to find the angle towards; note the parameter order is unusual by convention
      x - x-component of the point to find the angle towards; note the parameter order is unusual by convention
      Returns:
      the angle to the given point, in turns as a float; ranges from 0.0f to 1.0f
    • asin

      public static float asin(float a)
      Returns arcsine in radians; less accurate than Math.asin but may be faster. Average error of 0.000028447 radians (0.0016298931 degrees), largest error of 0.000067592 radians (0.0038727364 degrees). This implementation does not return NaN if given an out-of-range input (Math.asin does return NaN), unless the input is NaN.
      Parameters:
      a - asin is defined only when a is between -1f and 1f, inclusive
      Returns:
      between -HALF_PI and HALF_PI when a is in the defined range
    • asinDeg

      public static float asinDeg(float a)
      Returns arcsine in degrees. This implementation does not return NaN if given an out-of-range input (Math.asin does return NaN), unless the input is NaN.
      Parameters:
      a - asin is defined only when a is between -1f and 1f, inclusive
      Returns:
      between -90 and 90 when a is in the defined range
    • asinTurns

      public static float asinTurns(float a)
      Returns arcsine in turns. This implementation does not return NaN if given an out-of-range input (Math.asin does return NaN), unless the input is NaN. Note that unlike atan2Turns(float, float), this can return negative turn values.
      Parameters:
      a - asin is defined only when a is between -1f and 1f, inclusive
      Returns:
      between -0.25 and 0.25 when a is in the defined range
    • acos

      public static float acos(float a)
      Returns arccosine in radians; less accurate than Math.acos but may be faster. Average error of 0.00002845 radians (0.0016300649 degrees), largest error of 0.000067548 radians (0.0038702153 degrees). This implementation does not return NaN if given an out-of-range input (Math.acos does return NaN), unless the input is NaN.
      Parameters:
      a - acos is defined only when a is between -1f and 1f, inclusive
      Returns:
      between 0 and PI when a is in the defined range
    • acosDeg

      public static float acosDeg(float a)
      Returns arccosine in degrees. This implementation does not return NaN if given an out-of-range input (Math.acos does return NaN), unless the input is NaN.
      Parameters:
      a - acos is defined only when a is between -1f and 1f, inclusive
      Returns:
      between 0 and 180 when a is in the defined range
    • acosTurns

      public static float acosTurns(float a)
      Returns arccosine in turns. This implementation does not return NaN if given an out-of-range input (Math.acos does return NaN), unless the input is NaN.
      Parameters:
      a - acos is defined only when a is between -1f and 1f, inclusive
      Returns:
      between 0 and 0.5 when a is in the defined range
    • atan

      public static float atan(float i)
      Arc tangent approximation with very low error, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise). This method is usually about 4x faster than Math.atan(double), but is somewhat less precise than Math's implementation. For finite inputs only, you may get a tiny speedup by using atanUnchecked(double), but this method will be correct enough for infinite inputs, and atanUnchecked() will not be.
      Parameters:
      i - an input to the inverse tangent function; any float is accepted
      Returns:
      an output from the inverse tangent function in radians, from -HALF_PI to HALF_PI inclusive
      See Also:
    • atanDeg

      public static float atanDeg(float i)
      Arc tangent approximation returning a value measured in positive or negative degrees, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise). For finite inputs only, you may get a tiny speedup by using atanUncheckedDeg(double), but this method will be correct enough for infinite inputs, and atanUnchecked() will not be.
      Parameters:
      i - an input to the inverse tangent function; any float is accepted
      Returns:
      an output from the inverse tangent function in degrees, from -90 to 90 inclusive
      See Also:
    • atanTurns

      public static float atanTurns(float i)
      Arc tangent approximation with very low error, using an algorithm from the 1955 research study "Approximations for Digital Computers," by RAND Corporation (this is sheet 11's algorithm, which is the fourth-fastest and fourth-least precise). For finite inputs only, you may get a tiny speedup by using atanUncheckedTurns(double), but this method will be correct enough for infinite inputs, and atanUncheckedTurns() will not be.
      Parameters:
      i - an input to the inverse tangent function; any float is accepted
      Returns:
      an output from the inverse tangent function in turns, from -HALF_PI to HALF_PI inclusive
      See Also: