How to calculate an apparent dip from a real dip (and viceversa) using orthographic projection and trigonometry

Any student of geology in any university in the world learn during its degree the relationship between the real dip and the infinite apparent dips that a plane contains. Most of students learn how to calculate a real dip from a couple of apparent dips or, inversely, how to work out an apparent dip given the real dip and another direction using the stereonet. 

Stereographic projection provides an awesome graphic method for these calculation, which is very useful if one has to do a bunch of calculation, but it is not very useful if we have to deal with a large collection of data, or if we need to have a high degree of precision. Then, it is time for orthographic projections and the always useful trigonometry. 

Let's imagine that we have a known plane, and we need to calculate an apparent dip. We know the dip angle and the dip direction or strike, and obviously we know the direction along we want to know the apparent dip. In this context, 

δ = real dip. Note that the real dip is always measured along the maximum slope direction for a plane. No apparent dip can be larger than the real dip. 

α = apparent dip. This is the dip measure along a line which is not the maximum slope direction.

β = angle between the strike direction of the plane and the apparent dip direction.

You could think that it is difficult, but it is actually quite easy. The "trick" lies on relating the three triangles involved in the diagram (one containing a and b, another containing c and b, and another containing a and c). (Note that c is the hypotenuse of the horizontal triangle)

 The following trigonometric relations are quite straight and don't need much explanation:

(1)           sin β = a/c;  a = c ∙ sin β

(2)           tan δ = b/a; b = a ∙ tan δ

(3)           tan α = b/c; b = c ∙ tan α

(4)           b = a ∙ tan δ

(5)           b = c ∙ tan α

(6)           a = c ∙ sin β

Clear so far? Now, if you equal (4) and (5), and substitute a by (6),

(7)           a ∙ tan δ = c ∙ tan α

(8)           c ∙ sin β ∙ tan δ = c ∙ tan α

(9)           sin β ∙ tan δ = tan α

what you obtain is a direct relation between  α and δ. If you want to know the real dip from an apparent dip, use (11). If you want to calculate the apparent dip from the real dip, then use (10)

(10)          α = arctan (sin β ∙ tan δ)

(11)          δ =  arctan (tan α / sin β)

Easy, isn't it?

Why you would need to use that? Well, for example, I need it sometimes; I work interpreting satellite images, focusing on structural geology. When I measure fracture lengths on a plane, I cannot really measure their length: What I measure is an "apparent length". That means, I measure the projection of a line on a horizontal plane. For example, a 100 m fracture on a plane dipping 80 degrees will look very short if the direction of that fracture is the dip direction of the plane, but it will look as 100 m if the fracture is oriented along the strike. Any direction in between, will be variable. If it is variable, how can we correct it? knowing the apparent dip in that direction.

This method provides a way of correcting this distorsion, simply using any spreadsheet. You know the length of every single fracture, and the length you have measured. You also know the real dip of the plane (well, I can measure it on the DEM!), the strike, and that is all you need to know. But this will be another explanation, coming soon :-)

Feel free to make any comments, or perhaps any correction of suggestion. Hopefully this has been useful for you.