Delta robot - Inverse and forward kinematics library - (For @bitbeam bot)

ik.py

# Original code from
# http://forums.trossenrobotics.com/tutorials/introduction-129/delta-robot-kinematics-3276/
# License: MIT

import math

# Specific geometry for bitbeambot:
# http://flic.kr/p/cYaQah
e  =  26.0
f  =  69.0
re = 128.0
rf =  88.0

# Trigonometric constants
s      = 165*2
sqrt3  = math.sqrt(3.0)
pi     = 3.141592653
sin120 = sqrt3 / 2.0
cos120 = -0.5
tan60  = sqrt3
sin30  = 0.5
tan30  = 1.0 / sqrt3

# Forward kinematics: (theta1, theta2, theta3) -> (x0, y0, z0)
#   Returned {error code,theta1,theta2,theta3}
def forward(theta1, theta2, theta3):
    x0 = 0.0
    y0 = 0.0
    z0 = 0.0
    
    t = (f-e) * tan30 / 2.0
    dtr = pi / 180.0
    
    theta1 *= dtr
    theta2 *= dtr
    theta3 *= dtr
    
    y1 = -(t + rf*math.cos(theta1) )
    z1 = -rf * math.sin(theta1)
    
    y2 = (t + rf*math.cos(theta2)) * sin30
    x2 = y2 * tan60
    z2 = -rf * math.sin(theta2)
    
    y3 = (t + rf*math.cos(theta3)) * sin30
    x3 = -y3 * tan60
    z3 = -rf * math.sin(theta3)
    
    dnm = (y2-y1)*x3 - (y3-y1)*x2
    
    w1 = y1*y1 + z1*z1
    w2 = x2*x2 + y2*y2 + z2*z2
    w3 = x3*x3 + y3*y3 + z3*z3
    
    # x = (a1*z + b1)/dnm
    a1 = (z2-z1)*(y3-y1) - (z3-z1)*(y2-y1)
    b1= -( (w2-w1)*(y3-y1) - (w3-w1)*(y2-y1) ) / 2.0
    
    # y = (a2*z + b2)/dnm
    a2 = -(z2-z1)*x3 + (z3-z1)*x2
    b2 = ( (w2-w1)*x3 - (w3-w1)*x2) / 2.0
    
    # a*z^2 + b*z + c = 0
    a = a1*a1 + a2*a2 + dnm*dnm
    b = 2.0 * (a1*b1 + a2*(b2 - y1*dnm) - z1*dnm*dnm)
    c = (b2 - y1*dnm)*(b2 - y1*dnm) + b1*b1 + dnm*dnm*(z1*z1 - re*re)
    
    # discriminant
    d = b*b - 4.0*a*c
    if d < 0.0:
        return [1,0,0,0] # non-existing povar. return error,x,y,z
    
    z0 = -0.5*(b + math.sqrt(d)) / a
    x0 = (a1*z0 + b1) / dnm
    y0 = (a2*z0 + b2) / dnm

    return [0,x0,y0,z0]

# Inverse kinematics
# Helper functions, calculates angle theta1 (for YZ-pane)
def angle_yz(x0, y0, z0, theta=None):
    y1 = -0.5*0.57735*f # f/2 * tg 30
    y0 -= 0.5*0.57735*e # shift center to edge
    # z = a + b*y
    a = (x0*x0 + y0*y0 + z0*z0 + rf*rf - re*re - y1*y1) / (2.0*z0)
    b = (y1-y0) / z0

    # discriminant
    d = -(a + b*y1)*(a + b*y1) + rf*(b*b*rf + rf)
    if d<0:
        return [1,0] # non-existing povar.  return error, theta

    yj = (y1 - a*b - math.sqrt(d)) / (b*b + 1) # choosing outer povar
    zj = a + b*yj
    theta = math.atan(-zj / (y1-yj)) * 180.0 / pi + (180.0 if yj>y1 else 0.0)
    
    return [0,theta] # return error, theta

def inverse(x0, y0, z0):
    theta1 = 0
    theta2 = 0
    theta3 = 0
    status = angle_yz(x0,y0,z0)

    if status[0] == 0:
        theta1 = status[1]
        status = angle_yz(x0*cos120 + y0*sin120,
                                   y0*cos120-x0*sin120,
                                   z0,
                                   theta2)
    if status[0] == 0:
        theta2 = status[1]
        status = angle_yz(x0*cos120 - y0*sin120,
                                   y0*cos120 + x0*sin120,
                                   z0,
                                   theta3)
    theta3 = status[1]

    return [status[0],theta1,theta2,theta3]

This is very good code for delta thank you but İ cant understand the inverse kinematic equation " z = a + b*y " . How can you write this equation ??