Calculate the distance from a camera to a subject

tilt-distance.py

#!/usr/bin/python
# -*- coding: UTF-8 -*-
# 
# tilt-distance.py - calculate the horizontal distance to a target given camera pixel offset value
#
# To find d, the distance of the target from the camera, along the ground
# See https://drive.google.com/open?id=0B6e2rOpUwmteMWtOREdHSGdqOTg
#
# The triangle ΔABC is complementary to ΔDEC
#
# FOVp = the height of the camera field of view in pixels
# hT = the height of the the target (in feet or meters)
# ha = AC, the height of the target above the center line (feet or meters)
# hp = the image position of the target (in pixels)
# da = the distance of the target from the camera, along the center line
#
# We need to solve the 3 simultaneous equations
#
# 1) da = ha × FOVp ÷ (2 hp × tan(θ))
#
# 2) cos(α) = DE ÷ DC = d ÷ (da - BC)
#    → d = cos(α) × (da - BC) = cos(α) × (da - ha × tan(α))
#        = da cos(α) - ha sin(α)
#
# 3) tan(α) = EC ÷ DE = EC ÷ d
#    also, h = EC + AC = d × tan(α) + ha
#    → ha = h - d × tan(α)

import math

FOVp = 600
view = 25.5 * math.pi / 180.0       # angle of view converted to radians (θ in comments above)
hc = 1.0                            # camera height in feet
ht = (85) / 12.0 - hc               # 7'1" minus camera height
tilt = 9.7 * math.pi / 180.0        # tilt angle converted to radians (α in comments above)

def distance(hp):
    fov_adjustment = (FOVp * math.cos(tilt) / 2 / hp / math.tan(view)) - math.sin(tilt)
    return ht * fov_adjustment / (1 + fov_adjustment * math.tan(tilt))

for hp in range(10, 500, 10):
    print "hp=%d, distance=%3.1f feet" % (hp, distance(hp))