The Elements of Analytic Geometry

Couverture
 

Table des matières

Formulas and theorems from Trigonometry
19
Natural values of trigonometric functions
21
Rules for signs
22
CHAPTER II
23
Cartesian coördinates
24
Rectangular coördinates
25
Locus of a point satisfying a given condition
26
Angles
28
Orthogonal projection
29
Lengths
31
Principle of comparison
32
Inclination and slope
34
Point of division
38
Areas
42
Second theorem of projection
47
PAGE
51
The general equation of the circle
56
The general equation of the first degree Ax + By + C 0
85
Straight lines determined by two conditions
92
The normal form of the equation of the straight line
101
The angle which a line makes with a second line
109
The system of lines parallel to a given line
116
The parametric equations of the straight line
123
CHAPTER VI
149
Applications
156
Rotation of the axes
162
Application to equations of the first and second degrees
168
Transformation to rectangular coördinates
178
Conjugate hyperbolas and asymptotes
189
SECTION PAGE 78 Focal property of central conics
192
Types of loci of equations of the second degree
194
Construction of the locus of an equation of the second degree
197
CHAPTER XII
198
Systems of conics
200
CHAPTER IX
207
Equations of tangent and normal
210
Equations of tangents and normals to the conic sections
212
Tangents to a curve from a point not on the curve
215
Properties of tangents and normals to conics
217
Tangent to a curve at the origin
221
Second method of finding the equation of a tangent
223
CHAPTER X
226
Relative positions of lines of a system and a conic and of a line and conics of a system
228
Tangents to a conic
230
Tangent in terms of its slope
233
The equation in p
235
Tangents
237
Asymptotic directions and asymptotes
238
Centers
240
Diameters
241
Invariants under a translation of the axes
273
Nature of the locus of an equation of the second degree
275
Equal conics
278
Conics determined by five conditions
279
CHAPTER XIII
281
Translations
282
Displacements
284
The reflection in a line
287
Congruent and symmetrical conics
291
Similitude transformations
292
Similar conics
293
CHAPTER XIV
297
Inversion of conic sections
299
Angle formed by two circles
303
Angles invariant under inversion
304
Inversion of systems of straight lines
306
Inversion of a system of concentric circles
307
Orthogonal systems of circles
308
Construction of poles and polars
311
Polar reciprocation with respect to the locus of any equation
317
Correlations
323
Direction cosines of a line
330
CHAPTER XVII
338
Discussion of the equations of a curve Third fundamental
344
CHAPTER XVIII
348
The general equation of the first degree Ax + By + Cz + D 0
349
Planes determined by three conditions
353
The equation of a plane in terms of its intercepts
356
The angle between two planes
357
Systems of planes
359
CHAPTER XIX
363
The projecting planes of a line
366
Various forms of the equations of a straight line
369
Relative positions of a line and plane
373
Geometric interpretation of the solution of three equations of the first degree
374
CHAPTER XX
379
Cylinders
383
The projecting cylinders of a curve
384
Cones
385
Surfaces of revolution
386
Ruled surfaces
387
CHAPTER XXI
391
Polar coördinates
393
Spherical coördinates
394
CHAPTER XXII
397
The elliptic paraboloid
401
CHAPTER XXIII
410
Centers
419
Droits d'auteur

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Fréquemment cités

Page 417 - The locus of the middle points of a system of parallel chords in a parabola is called a diameter.
Page 97 - The line joining the middle points of two sides of a triangle is parallel to the third side and equal to half of the third side.
Page 64 - A point moves so that the sum of the squares of its distances from the points (0, 0), (1, 0) is constant.
Page 63 - Find the equation of the locus of a point which moves so that its distances from (8, 0) and (2, 0) are always in a constant ratio equal to 2.
Page 27 - The projection of a point upon a line is the foot of the perpendicular from the point to the line. 329. DEF. The projection of one line upon another is the segment between the projections of the extremities of the first line upon the second. A' / ri U/ A A B' A
Page 64 - A point moves so that the difference of the squares of its distances from two fixed points is constant. Show that...
Page 333 - ... cos y cos y' = 0. That is, two lines are parallel and in the same direction when and only when their direction angles are equal, and perpendicular when and only when the sum of the products of their direction cosines is zero.
Page 85 - N6 is to say that if two nonvertical lines are perpendicular, then the slope of one is the negative reciprocal of the slope of the other.
Page 55 - Find the equations of the perpendicular bisectors of the sides of the triangle in problem 2, and show that they meet in a point.
Page 74 - A conic section is the locus of a point whose distances from a fixed point and a fixed line are in a constant ratio. 4. Show that every conic is represented by an equation of the second degree in x and y. Hint. Take Y Y' to coincide with the fixed line, and draw XX

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