Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green
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|What are the equations of lines AB and OP?. The generic equation of a
line is y = mx + b, where b is the y-intercept and m is the slope. For
line AB, the slope is -tan q, or -b/a. Thus we have:|
Line AB: y = -bx/a + b, or, x/a + y/b = 1.
If OP is perpendicular to AB, its y- intercept is obviously zero, and we have y = x tan p, or y = xa/b, or xa = yb.
Thus, a general line has a simple equation in terms of its intercepts, and a line through the origin perpendicular to the line also has a simple equation in terms of the intercepts. As a general rule, if two lines are perpendicular, the product of their slopes is -1.
|The equation for a plane in three dimensions is exactly analogous for
the line in two dimensions: x/a + y/b + z/c = 1.
If line OP is perpendicular to the plane, its equation is also analogous to the two-dimensional case: xa = yb = zc.
If we know a plane, we know the line through the origin perpendicular to it, called the pole to the plane. Quantities a, b and c are termed the direction numbers of the line.
Line OP makes angles A, B, and C with the axes. The angles are related as follows: cos2 A + cos2 B + cos2 C = 1
Cos A, cos B and cos C are called the direction cosines of the line. They are obviously also direction numbers. If you know any arbitrary direction numbers a, b and c, then cos A = a/s, cos B = b/s and cos c = c/s, where s2 = a2 + b2 + c2
Now that we know the equation of a plane in space, the rules for Miller Indices are a little more intelligible. They are:
For example, if the x-, y-, and z- intercepts are 2,1, and 3, the Miller indices are calculated as:
Thus, the Miller indices are 3,6,2. If a plane is parallel to an axis, its intercept is at infinity and its Miller index is zero. A generic Miller index is denoted by (hkl).
If a plane has negative intercept, the negative number is denoted by a bar above the number. Never alter negative numbers. For example, do not divide -1, -1, -1 by -1 to get 1,1,1. This implies symmetry that the crystal may not have!
For hexagonal and trigonal minerals, there are three possible axis directions, spaced 120 degrees apart:
+x2\ /-x3 \ / \ / \ / \ / -x1 \ / +x1 ----------------+---------------- / \ / \ / \ / \ / \ +x3/ \-x2
Obviously, any two intercepts specify the face. Also, there will be two intercepts of one sign and one of the other. The Miller indices for a hexagonal mineral are often written hikl. Indices h, i and k are related by h + i + k = 0. Some modern texts dispense with the i term and treat hexagonal minerals like all others.
The Miller Index of a line is about as simple as it can be: if the line passes through (h, k, l), its Miller Index is [hkl], written in brackets to distinguish it from a face.
A family of faces all parallel to some particular line is called a zone, and the line is called the zone axis. Two faces (hkl) and (pqr) belong to zone [kr-lq, lp-hr, hq-kp]. Note the similarity of this formula to the cross-product formula from vector mechanics. Any other face (def) belongs to the same zone if its indices are some linear combination of (hkl) and (pqr), for example, d = 2h-3p, e = 2k-3q, etc.
For example, faces (110) and (010) belong to [1*0-0*1, 0*0-1*0, 1*1-1*0], or . The final zero in the face indices is a tipoff that they are both parallel to the z-axis, and the zone index  is the z-axis. Any other face whose index is some linear combination of (110) and (010) is also a member of that zone. Obviously the final index must be zero.
What about faces (211) and (124)? Their zone axis is [1*4-1*2, 1*1-2*4, 2*2-1*1] or [2,-7,1]. Faces (335), (546), (1,-1,-3), etc. also belong to this zone.
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Created 9 Oct 1997, Last Update 14 Oct 1997
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