Geomathematics. Mathematical background and geo-science by F. P. Agterberg

By F. P. Agterberg

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11] where : 0 —r /2x 2 From comparing this result with the expression y - a + bx, it follows that tan a = -lib. The constant d0 represents the perpendicular distance from the origin to the line. 6 DIFFERENTIAL CALCULUS 37 When b = c = 0, this expression reduces to y as quadratic polynomial in x. If 0=cand Z? = 0, the curve satisfying the equation is a circle which is a special form of the ellipse. 8. 12] b2 > Aacy or a hyperbola if b2 < 4ac. In the special case b2 = Aacy the curve is a parabola whose axis of symmetry is not necessarily parallel to the F-axis as when b = c = 0.

The angle between sides a and b is 90°. Some definitions are : sine of 0 = sin 0 = a/c cosine of 0 = cos 0 - b/c tangent of 0 = tan Θ = a/b Also: Θ = arcsin (a/c) = arccos (b/c) = arctan (a/b) This means that Θ is the angle (subtending an arc on a circle) for which the sine is equal to the ratio a/c, etc. The angle 0 can be represented by using radians or by using degrees. 01745 radians From the definitions, it follows that : a = c sin Θ ; b = c cos Θ The Pythagorean theorem then becomes: c2 = a2 + b2 = c 2 (sin 2 0 + cos2 Θ) and: sin20 + cos20 = l Suppose that P} and P2 are two points in the plane with coordinates P.

The angle of slope is 45°. One can construct functions JC2, x , . . , xn where « is a positive integer. If cQ, cv c2,... ,c denote arbitrary constants: f{x) = cQ + c^x + c2x + . . + cnxn denotes a function that is called a polynomial in x of degree n. It can be said that/(jc) is a linear combination of the functions 1, x, x2,... xn. , /(*) = 1 + 2x. Polynomials of the second, third, fourth, fifth and sixth degree are called quadratic, cubic, quartic, quintic, and sextic, respectively. 6A). If φ is the angle of slope, then b = tan φ.