Notes On Elements of Analytical Solid Geometry
Notes On Elements of Analytical Solid Geometry
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Also, let MN = x, MP = y, ON 2, be the coordinates of any point P in the generating ellipse ; then the ellipse PQR gives "NQ 2 + NlT 2 = I NQ 2 z* Also from hyperbola AQ 2 "~=i.
NR 2 z" And from hyperbola BR -- ^ i.
Hence, f A 8 V 2 Z 2 or - + ^- - i (102) the equation to the surface.
92. To determine the form of the hyperboloid of one sheet from its equation.
Since the equation (102) admits values of x, y and z positive and negative however large, the surface is extended indefinitely on all si...des of the origin. If we put z = o we obtain ~z- + ~i = i for the trace on xy which is the ellipse AB. Similarly r 2 s 2 the sections by the planes xz and j 2 are respectively ^ - i the v 2 z hyperbola AQ, and 1 -^ ^ = i the hyperbola BR. The ellipse AB and the hyperbolas AQ and BR are the principal sections. The sections parallel to xy are all ellipses similar to and greater than AB. The sections parallel to xz and yz are hyperbolas similar to the principal sections.
The semi-axes a and b are called the real semi-axes of the surface and c the imaginary semi-axis, since x = o and y = o give z = c\/ i.
What to read after Notes On Elements of Analytical Solid Geometry?
You can find similar books in the "Read Also" column, or choose other free books by Charles S Charles Scott Venable to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
Also, let MN = x, MP = y, ON 2, be the coordinates of any point P in the generating ellipse ; then the ellipse PQR gives "NQ 2 + NlT 2 = I NQ 2 z* Also from hyperbola AQ 2 "~=i.
NR 2 z" And from hyperbola BR -- ^ i.
Hence, f A 8 V 2 Z 2 or - + ^- - i (102) the equation to the surface.
92. To determine the form of the hyperboloid of one sheet from its equation.
Since the equation (102) admits values of x, y and z positive and negative however large, the surface is extended indefinitely on all si...des of the origin. If we put z = o we obtain ~z- + ~i = i for the trace on xy which is the ellipse AB. Similarly r 2 s 2 the sections by the planes xz and j 2 are respectively ^ - i the v 2 z hyperbola AQ, and 1 -^ ^ = i the hyperbola BR. The ellipse AB and the hyperbolas AQ and BR are the principal sections. The sections parallel to xy are all ellipses similar to and greater than AB. The sections parallel to xz and yz are hyperbolas similar to the principal sections.
The semi-axes a and b are called the real semi-axes of the surface and c the imaginary semi-axis, since x = o and y = o give z = c\/ i.
What to read after Notes On Elements of Analytical Solid Geometry?
You can find similar books in the "Read Also" column, or choose other free books by Charles S Charles Scott Venable to read onlineMoreLess
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