You should be able to:

1. measure the height of a object from its shadow

2. measure the height of an object using its vertical displacement

A. geometry of a vertical photo

1. the principle point is the center of the image

a. determined using the fiducial marks on the sides and corners of the photo

b. can only be accurately determined if you have the entire photo

c. can sometimes be estimated since vertical displacement causes tall objects to appear to lean away from the nadir

2. nadir is the point directly below the camera

a. most measurements assume the principle point is on the nadir

b. the nadir and the principle point are the same unless the image is tilted

c. most measurements on a photo will be okay unless the camera is tilted more than 3 deg.

3. any vertical image creates vertical displacement or vertical parallax. Any part of an image that is above another will be displaced away from the nadir relative to those below it, and conversely anything that is at a lower elevation will be displaced toward the center relative to those above it.

a. the amount of parallax increases as you move away from the nadir on the image

b. the amount of parallax increases with the height of the object

c. the amount of parallax decreases with the height of the camera

i. note that this is independent of scale. a photo taken at 1000 ft with a 3” lens will have the same scale as a photo taken at 2000 ft with a 6” lens (1:4000) but the image from 2000 ft will have less parallax

B. there are two ways to make a height measurement on a single photo. Both only work for a vertical object such as a building, tree or pole.

1. shadow length

a. if you know the date, time, scale and location of the photo the height of an object can be calculated from h = s * tan(theta)

i. s is the length of the shadow in the real world (determined by measuring the length of the shadow on the image and using the scale)

ii. theta is the angle of the sun above the horizon determined by the date and time from a table of sun altitudes (such as aa.usno.navy.mil/data/docs/AltAz.html)

iii. the units of the answer are the same as the units of the shadow length

b. if you can identify a north south line in the image, you can determine the time from the angle of the shadows relative to the line.

c. also remember that you can compare the shadow length to the shadow of an object of known height. (especially good for getting a sense of relative height, e.g. taller than the trees or bigger than a house…)

2. parallax displacement

a. if the top and bottom of a object are visible and the object is far enough from the center to show vertical displacement, the height can be calculated from h=H * d/r

i. H = height of the camera above the ground at the bottom of the object

ii. r = measurement from the principle point to the top of the object

iii. d= measurement from the top of the object to the bottom along the line from the principle point

iv. points to remember

- r and d must be in the same units (but H does not)

- in hilly terrain, H may not be the given H of the photograph

- units of h are the same as the units of H

- scale of the photo is not needed