Section 1: Problem 4 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Make a detailed comparison of the affine conics in (1.3) with the projective conics in (1.6).

Affine conics correspond to projective conics in the usual way: the equation of an affine conic $\begin{eqnarray*} ax^{2}+bxy+cy^{2}+dx+ey+f & = & 0 \end{eqnarray*}$ is considered as the intersection of a projective conic with $z=1$ , and the corresponding equation of the projective conic is then $\begin{eqnarray*} aX^{2}+bXY+cY^{2}+dXZ+eYZ+fZ^{2} & = & 0. \end{eqnarray*}$

Below is the table of all affine conics and corresponding projective conics.

Case	Equation	Homogeneous equation*	Case
(a) ellipse	$x^{2}+y^{2}-1=0$	$x^{2}+y^{2}-z^{2}=0$	( $\alpha$ ) nondegenerate conic
(b) parabola	$x^{2}-y=0$	$x^{2}-y^{2}+z^{2}=0$	( $\alpha$ ) nondegenerate conic
(c) hyperbola	$x^{2}-y^{2}-1=0$	$-x^{2}+y^{2}+z^{2}=0$	( $\alpha$ ) nondegenerate conic
(d) single point	$x^{2}+y^{2}=0$	$x^{2}+y^{2}=0$	( $\delta$ ) one point
(e) empty set	$x^{2}+y^{2}+1=0$	$x^{2}+y^{2}+z^{2}=0$	( $\beta$ ) empty set
(f) empty set	$x^{2}+1=0$	$x^{2}+z^{2}=0$	( $\delta$ ) one point
(g) empty set	$1=0$	$z^{2}=0$	( $\epsilon$ ) double line
(h) line	$x=0$	$x^{2}-z^{2}=0$	( $\gamma$ ) line pair
(i) line pair	$xy=0$	$x^{2}-y^{2}=0$	( $\gamma$ ) line pair
(j) parallel lines	$x^{2}-x=0$	$x^{2}-z^{2}=0$	( $\gamma$ ) line pair
(k) double line	$x^{2}=0$	$x^{2}=0$	( $\epsilon$ ) double line
(l) whole plane	$0=0$	$0=0$	( $\zeta$ ) whole space

* In suitable coordinates.

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Section 1: Problem 4 Solution

Section 1: Problem 4 Solution