Useful Geometrical Facts to Know

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Construction Errors

Even the best draftsman will make unconscious errors, like habitually measuring angles slightly too large or too small, or drawing lines systematically to the left or right of a point the line is supposed to pass through. Also, geologic data have inaccuracies of measurement, and geologic structures are rarely geometrically simple. So constructions will often have slight errors in them even with the most careful work.

Nowadays, geologists are as likely to plot drawings on a computer as on paper. Nevertheless, the same geometrical principles apply. The accuracy of the construction is only as good as the computer input, and that in turn is only as good as the field measurements. So errors still occur and have to be dealt with.

 Many constructions depend on finding the intersections of lines. A small error in the location of one line will result in a large error in the position of the point when the lines meet at a small angle. Try to plan constructions so that lines meet at large angles.

The larger you make your diagram, the less serious drafting errors will be. Drafting errors tend to be pretty much constant in size. An error of 0.5 mm in a line 1 cm long is a 5% error; the same error of 0.5 mm in a line 10 cm long is an 0.5% error.

Try to interpolate rather than extrapolate. Interpolation, or estimating values between two or more known values, can be done with fairly high confidence. The errors are usually no greater than the errors in the data. If you extrapolate, or estimate values beyond the range of known data, any errors in the data will cause increasingly large errors the further you extrapolate. In addition, you have no way of knowing whether the pattern you are extrapolating might change; for example, extrapolating the depth of a bed from known data will not alert you to the presence of a fault.

 Often three lines should ideally intersect at a point, but do not meet in practice. Review the construction for errors. If the remaining residual error is small, and the error triangle is more or less equilateral, the best estimate for the location of the point will likely be the center of the error triangle. If the error triangle is long and thin, the midpoint of the shortest side is probably the best estimate.

DO NOT CONSIDER THESE HINTS AS A LICENSE TO BE CASUAL OR SLOPPY IN YOUR WORK! Preventing errors is always better than correcting for them.

Necessary Data to Perform Constructions

Lines and Planes

 Intersecting straight lines determine a point Two points on a line determine a line Three points on a plane determine a plane Two intersecting lines on a plane determine a plane A line and a point not on the line determine a plane The intersection of a line and a plane is a point (unless the line and plane are parallel or coincide) The intersection of two planes is a line (unless the planes are parallel or coincide) The intersection of three planes is a point (unless some planes are parallel or coincide)

Circles and Spheres

 Intersecting circles may determine two, one, or zero points The center of a circle and a point on its circumference determine a circle The diameter of a circle determines the circle Three points on the circumference of a circle determine a circle The center of a sphere and a point on its circumference determine a sphere Four points on the circumference of a sphere determine a sphere. The sphere is found more easily by computation than construction The intersection of a plane and a sphere is always a circle.

Ellipses and Ellipsoids

 Five points determine any conic section (circle, ellipse, parabola or hyperbola) but the constructions are very complex. You're better off determining the curve mathematically. The major and minor axes of an ellipse determine the ellipse. The major axis is the longest diameter of the ellipse; the minor axis is the shortest. If the two axes are equal, the ellipse is a circle. Circles are special cases of ellipses. The easiest way to construct an ellipse is the trammel method. Let the major axis be a and the minor axis be b. Construct the major and minor axes at the desired location. Next, mark points O, A, and B on the edge of a piece of paper. Let OA = a and OB = b. Slide the paper so that A moves along the minor axis and B on the major axis. Point O then traces out the ellipse. The principal axes of an ellipsoid determine the ellipsoid. These are the longest and shortest diameters of the ellipsoid and the diameter at right angles to the long and short axes. The principal sections of an ellipsoid determine the ellipsoid. These are the planes containing each pair of principal axes. Although the construction is complex, any three sections of an ellipsoid determine the ellipsoid. This fact is important in determining strain from deformed structures. It's better to use a computer to find the ellipsoid mathematically. The intersection of a plane and an ellipsoid is always an ellipse (the ellipse can be a circle.)

Triangles

 The three sides of a triangle determine the triangle One side of a triangle and the two adjacent angles determine a triangle. This fact is the basis for triangulation. Two sides of a triangle and the included angle determine a triangle Two sides of a triangle and a non-included angle allow two possible triangles Known: AC, BC, angle A. B and B' are possible locations for vertex B The altitudes of a triangle meet at a common point. If the triangle is obtuse, the point can lie outside the triangle. The medians of a triangle (lines from a vertex to the midpoint of the opposite side) meet at a common point (the centroid) AO = 2/3 AX If three lines that should intersect don't, but enclose a triangle instead, its centroid is the best estimate of the true intersection. The angle bisectors of a triangle meet at a common point. A circle tangent to all three sides has its center at this point. The circle is the incircle and the point is the incenter The perpendicular bisectors of the sides of a triangle meet at a common point. The circle that passes through all three vertices (the circumcircle) has its center at this point (the circumcenter) The circumcenter need not lie within the triangle. We can use this fact to find the center of an unknown circle given three points on its circumference.

Parallel Lines

 Vertical angles: -- Angle A = D -- Angle B = C -- Angle A + C = 180 -- Angle B + D = 180 Parallel lines-- All angles A are equal -- All angles B are equal -- A + B = 180 Equally-spaced parallel lines divide any intersecting line into equal segments AB = BC = CD We can use the previous fact to subdivide any line segment into equal segments. To subdivide AB, construct an arbitrary line AC and lay off the desired number of equal divisions on it. Construct CB and the parallel lines as shown.

Angles

 To bisect an angle, set your compass at the vertex of the angle and draw an arc A. Then set the compass where A cuts the angle (B and C) and draw two more arcs (E and F) These meet at G. A line through G and V bisects the angle. To construct the perpendicular bisector of a line segment, set the radius of your compass greater than half the length of the line. Set the compass at each end of the line and draw two arcs A and B, which intersect at C and D. Line CD is the perpendicular bisector. The radius of a circle is always perpendicular to the tangent at the point where the radius meets the circle The perpendicular bisector of a chord of a circle also bisects the arc enclosed by the chord and passes through the center of the circle. The preceding fact suggests how to find the center of a circle given three points on the circumference A, B, and C: Construct chords AB, AC, and BC Construct the perpendicular bisectors to AB, AC and BC. They intersect at the center. (Actually you only need two, but the third serves as a check). An inscribed angle in a circle is equal to half the central angle. This is true no matter where the vertex of the inscribed angle is. This fact is one of the least-known and most useful geometric facts. As a consequence of the above, any angle inscribed in a semicircle is a right angle.

Congruent and Similar Figures

Congruent Figures

Figures that are identical in all their measurements are congruent. If you can prove that two figures are congruent, you can use measurements from one to determine dimensions of the other. The following are congruent figures:

• All circles of equal radius or diameter
• Ellipses with the same major and minor axes
• Triangles with one side and two adjacent angles equal
• Triangles with two sides and the included angle equal
• Triangles with all three sides equal
• Regular polygons with equal numbers of sides and any side equal

Similar Figures

Figures that have corresponding angles equal and all dimensions in a given ratio are similar. If you can show that two figures are similar, and can find the proportion between them, you can use any dimension from one figure to find the corresponding dimension on the other. The following are similar:

• All circles are similar
• All parabolas are similar
• All regular polygons with equal numbers of sides are similar.
• Triangles with two equal angles are similar
• Right triangles with one other equal angle are similar
• Isoceles triangles with equal apical or base angles are similar

These two special cases are very common:

• The angles of a triangle sum to 180 degrees
• The angles of an N-sided polygon sum to (N-2)180 degrees. If n=3, the sum is 180 degrees

An Important Triangle Construction

This construction will crop up a number of times in a variety of applications, so it is useful to learn it. Lines AB and OX are perpendicular. It is easy to see that triangles AOB, BCO, and ACO are similar; that is, all their angles are equal. Therefore all their sides have constant ratios.

 Call: a = OB b = OA c = BC d= AC e = OC f = AB = c+d

We can see that

• b/f = e/a =====> e = ab/f
• a/b = c/e =====> c = ae/b = a(ab/f)/b2 = a/f
• d/b = e/a =====> d = eb/a = (ab/f)b/a2 = b /f

From the Pythagorean Theorem, note that  f2 = a2 + b2  or f = √(a2 + b2 )

and

c = a2/√(a2 + b2 ), d = b2/√(a2 + b2 ), e = ab/√(a2 + b2 )

Notes for Computer Users

It is tempting to think that in this day of microcomputers and sophisticated graphics that geometrical techniques are obsolete in structural geology or soon will be. Far from it!

A thorough knowledge of geometry is absolutely necessary for computer graphics and writing structural geology programs on a computer. In fact, the need is even more stringent than ever. A high-school drafting course would be enough geometrical background to pass a pre-computer course in structural geology. You cannot program a computer or calculator to solve structural problems without a good knowledge of three-dimensional analytical geometry, a subject normally encountered in the second year of college calculus. It is often possible to do graphical constructions without knowing trigonometry; it is absolutely impossible to program computers to solve structural problems without a thorough knowledge of trigonometry.

Many students expect to get a lucrative job programming computers. There are relatively few really basic, widely-used utility programs and these have all pretty much been written. The future belongs to people who can program computers for specific applications and use existing software to solve problems. You cannot program a computer unless you can solve the problem yourself.

Whether geologic information is published in print or displayed on a computer terminal screen, various sorts of graphical methods will have to be used. A geologist must be able to read and interpret a wide variety of diagrams. This need will only increase as graphics systems become more widespread and sophisticated.

The vastly increased computing power available to geologists makes it possible to solve problems that were out of reach a few years ago. Not only do the methods of solving these problems use geometry, but it is often far more sophisticated than structural geologists of yore even dreamed of.

It will be some time yet before geologists can take computers in the field that are small enough and rugged enough to use and powerful enough to solve structural problems on the spot (although palmtop computers are fast approaching that state). The only alternative in such situations is to be able to solve the problems manually.

Created 5 January 1999, Last Update 30 January 2012
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