A Treatise On the Analytic Geometry of Three Dimensions 1
A Treatise On the Analytic Geometry of Three Dimensions 1
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The following examples will serve further to illustrate the principles which have been laid down : Ex. 1. To find the locus of the intersection of generaton to a hyperboloid which cat at right angles.
The section parallel to the tangent plane which contains the generaton most be an equilateral hyperbola, so that (Art. 164) (a"«-a'*) + (a"«-o'"»)=0. But (Art. 161) the square of the radius vector to the point is a"s+6"2+c"«-(o"«-a'2)-(a"«-o'"«).
We hare, therefore, the locus a sphere, the square ...of whose radios is equal to a"^-\-V*^+c^\ Otherwise thus: If two generators are at right angles, their plane together with the plane of each and of the normal at the point, are a system of three tangent planes to the surface, mutually at right angles, whose intersection lies on the sphere r«=a"»+6"«+c"« (Art. 93).
Ex. 2. To find the locus of the intersection of three tangent lines to a qoadzio mutually at right angles (see Ex. 6, Art. 121).
Let a, /9, 7 be the angles made by one of these tangents with the normals thioogh the locus point, and since each of these tangents lies in the tangent cone thxou^ that point, we hare the conditions cos'g cos*^ ootf y _ oos'g' c oe*/y oos'y^ _ »'«-.«« a"^-'a^ a'"*-a^ ~ ' coB» a" cos*jy' COS* y" __ FOCAL CONICS AND CX)NFOCAL SURFACES.
What to read after A Treatise On the Analytic Geometry of Three Dimensions 1?
You can find similar books in the "Read Also" column, or choose other free books by George Salmon to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
The following examples will serve further to illustrate the principles which have been laid down : Ex. 1. To find the locus of the intersection of generaton to a hyperboloid which cat at right angles.
The section parallel to the tangent plane which contains the generaton most be an equilateral hyperbola, so that (Art. 164) (a"«-a'*) + (a"«-o'"»)=0. But (Art. 161) the square of the radius vector to the point is a"s+6"2+c"«-(o"«-a'2)-(a"«-o'"«).
We hare, therefore, the locus a sphere, the square ...of whose radios is equal to a"^-\-V*^+c^\ Otherwise thus: If two generators are at right angles, their plane together with the plane of each and of the normal at the point, are a system of three tangent planes to the surface, mutually at right angles, whose intersection lies on the sphere r«=a"»+6"«+c"« (Art. 93).
Ex. 2. To find the locus of the intersection of three tangent lines to a qoadzio mutually at right angles (see Ex. 6, Art. 121).
Let a, /9, 7 be the angles made by one of these tangents with the normals thioogh the locus point, and since each of these tangents lies in the tangent cone thxou^ that point, we hare the conditions cos'g cos*^ ootf y _ oos'g' c oe*/y oos'y^ _ »'«-.«« a"^-'a^ a'"*-a^ ~ ' coB» a" cos*jy' COS* y" __ FOCAL CONICS AND CX)NFOCAL SURFACES.
What to read after A Treatise On the Analytic Geometry of Three Dimensions 1?
You can find similar books in the "Read Also" column, or choose other free books by George Salmon to read onlineMoreLess
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