sinusoidal_monotonic_curve.py

# -*- coding: utf-8 -*-

"""
Monotonic increasing sinusoidal curve with same entrance/exit angle
"""
import numpy as np
import pylab


def discrt_tan(x0, y0, x1, y1):
    """ discrete tangent function for tuple of points """
    return np.divide(np.subtract(y1,y0), np.subtract(x1,x0))


def curve(xs, size, mid_angle=0.0):
    """ monotonic raising curve function with sinusoidal accent.
    
    Function for a monotonic raising curve with same entrance/exit angle.
    Sinusoidal setup for *monotonic increasing* constraints angles in following fashion;
        - mid_angle must be between [0..pi/4) - approaching pi/4 yields straight line.
        - generated entrance/exit angles are between [45..~63.43] degrees
    
    The curve is expresses as;
        y = x + [sin(2PI * x / size)] / accent    

    Args:
        xs (array like): input value(s)
        size: size range of the input/output values
        mid_angle: middle angle of the sinusoidal in radians
    
    Returns:
        array-like curve value.

    Raises:
        `ValueError` for params. that out of range;
            mid_angle - must be between [0..pi/4)
            xs - must be in range between [0..size]

    Todo:    
        - `accent` does not well-define middle angle parameter.
        - if curve will not be limited in size range, derivation will be step-wise continuous.
            More calculation can be made for continuous derivation. (tangents at 0 and size)
        
    """
    if mid_angle<0 or mid_angle>=np.pi/4:
        raise ValueError("mid-angle must be between [0..pi/4)")

    if np.any(xs < 0) or np.any(xs > size):
        raise ValueError("xs must be between [0..size]")

    accent = 2*np.pi/size * np.tan(np.pi/4+mid_angle)  # TODO: not well-defined mid-angle param.
    return xs + np.sin(2*np.pi*xs/size) / accent


if __name__ == "__main__":
    """ 
    Test and plot `curve` function for various values and mid_angle degrees
    """
    eps = 1e-11
    #eps = np.finfo(float).eps
    size = 1

    print "### testing values..."
    print "x= {:.15f}, y={:.15f}".format(size, curve(size, size))
    print "x= {:.15f}, y={:.15f}".format(np.subtract(size,eps), curve(np.subtract(size,eps), size))
    print "x= {:.15f}, y={:.15f}".format(size/2.0, curve(size/2.0, size))

    # angles at 0, mid, size
    mids = [0, np.pi/128, np.pi/64, np.pi/32, np.pi/16, np.pi/8, np.pi/6, np.pi/4-eps]
    max_err = 0.0

    params = {'legend.fontsize': 'small',
         'axes.labelsize': 'small',
         'axes.titlesize':'small',
         'xtick.labelsize':'small',
         'ytick.labelsize':'small'}
    pylab.rcParams.update(params)
    colc = 2
    rowc = (len(mids)+1)/colc
    f, axarr = pylab.subplots(rowc, colc)

    print "\n### testing angles..."

    for i,mid in enumerate(mids):
        print "mid (input)   = {:.15f}".format(np.rad2deg(mid))
        t = discrt_tan(size/2.0, curve(size/2.0, size, mid),
                np.add(size/2.0,eps), curve(np.add(size/2.0,eps), size, mid))
        print "angle at mid  = {:.15f}".format(np.rad2deg(np.arctan(t)))

        t = discrt_tan(0, curve(0, size, mid),
                eps, curve(eps, size, mid))
        print "angle at 0    = {:.15f}".format(np.rad2deg(np.arctan(t)))
        
        t1 = discrt_tan(size, curve(size, size, mid),
                np.subtract(size,eps), curve(np.subtract(size,eps), size, mid))
        print "angle at size = {:.15f}".format(np.rad2deg(np.arctan(t1)))
        
        # error
        err = np.abs(np.subtract(np.rad2deg(np.arctan(t1)), np.rad2deg(np.arctan(t))))
        max_err = np.maximum(max_err, err)
        print "error         = {:.15f}\n".format(err)

        # plot
        xs = np.linspace(0, size, 100)
        ys = curve(xs, size, mid)
        row = i/colc
        col = i - row*colc

        axarr[row, col].plot(xs, ys)
        axarr[row, col].set_title('mid_angle={:.4f} rad.'.format(mid))
        axarr[row, col].text(0.95, 0.15, 'alpha={:.6f} deg.'.format(np.rad2deg(np.arctan(t))),
            verticalalignment='bottom', horizontalalignment='right',
            transform=axarr[row, col].transAxes,
            color='green', fontsize=10)
        axarr[row, col].text(0.95, 0.03, 'err={:.6f} deg.'.format(err),
            verticalalignment='bottom', horizontalalignment='right',
            transform=axarr[row, col].transAxes,
            color='red', fontsize=10)

    print"Max. Error    = {:.15f}".format(max_err)
    pylab.show()