tailieunhanh - Handbook of mathematics for engineers and scienteists part 26

A right triangle is a triangle with a right angle. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs (Fig. . The hypotenuse c. the legs a and fr,and the angles a and t3 opposite the legs satisfy the following relations | . Quadric Surfaces Quadrics 143 The distance between them can be calculated by the formula ri - F2 RiR2 d --------------- R1X R2 X1 - X2 y1 - y2 Z1 - Z2 11 m1 n1 l2 m2 n2 1 mi I2 I mi m I2 I ni h I2 2 m2 m2 n2 n2 I2 The condition that the determinant in the numerator in is zero is the condition for the two lines in space to meet. Remark 1. The numerator of the fraction in is the volume of the parallelepiped spanned by the vectors ri - r2 Ri and R2 while the denominator of the fraction is the area of its base. Hence the fraction itself is the altitude d of this parallelepiped. Remark 2. If the lines are parallel . 11 l2 l m1 m2 m and n1 n2 n or R1 R2 R then the distance between them should be calculated by formula with ro replaced by r2. . Quadric Surfaces Quadrics . Quadrics Canonical Equations . Central surfaces. A segment joining two points of a surface is called a chord. If there exists a point in space not necessarily lying on the surface that bisects all chords passing through it then the surface is said to be central and the point is called the center of the surface. The equations listed below in Paragraphs to for central surfaces are given in canonical form . the center of a surface is at the origin and the surface symmetry axes are the coordinate axes. Moreover the coordinate planes are symmetry planes. . Ellipsoid. An ellipsoid is a surface defined by the equation x2 y2 z2 a2 b2 b2 1 where the numbers a b and c are the lengths of the segments called the semiaxes of the ellipsoid see Fig . Figure . Triaxial ellipsoid a and spheroid b . 144 Analytic Geometry If a b c then the ellipsoid is said to be triaxial or scalene. If a b c then the ellipsoid is called a spheroid it can be obtained by rotating the ellipse x2 a2 z2 2 1 y 0 lying in the plane OXZ about the axis OZ see Fig. . If a b c then the ellipsoid is an oblate spheroid and if a b c then the .