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Your trouble arises because Thurston assumes some familiarity with classical projective geometry. Put n=3 for example. Consider $\mathbb{R}^3\subset\mathbb{RP}^3$ as an affine chart, say, the chart $\{[1:x_1:x_2:x_3]\mid x_i\in\mathbb{R}\}$. The unit sphere is $$ S=\{[1:x_1:x_2:x_3]\mid x_1^2+x_2^2+x_3^2=1\}=\{[x_0:x_1:x_2:x_3]\mid x_1^2+x_2^2+x_3^2=x_0^2\}, $$ while the paraboloid is $$P=\{[1:x_1:x_2:x_3]\mid x_3=x_1^2+x_2^2\}=\{[x_0:x_1:x_2:x_3]\mid x_0x_3=x_1^2+x_2^2,\ x_0\neq 0\}.$$ $P$ has a "point at infinity" $[1:0:0:0]$. What Thurston really means by "paraboloid" when saying "maps the unit sphere to paraboloid" is P with this point at infinity attached, i.e. $\{[x_0:x_1:x_2:x_3]\mid x_0x_3=x_1^2+x_2^2\}$. To find the required map, try something like $[x_0:x_1:x_2:x_3]\mapsto [x_0+x_3:x_1:x_2:x_0-x_3]$.

The point is to understand that all conic hypersurfaces in ℝℙn are projectively equivalent (in other words, in projective geometry there is no distinction among ellipsoid, paraboloid and hyperboloid), and such a hypersurface delimits a convex body which serves as a model of $\mathbb{H}^n$, where geodesics are straight lines.

Your trouble arises because Thurston assumes some familiarity with classical projective geometry. Put n=3 for example. Consider $\mathbb{R}^3\subset\mathbb{RP}^3$ as an affine chart, say, the chart $\{[1:x_1:x_2:x_3]\mid x_i\in\mathbb{R}\}$. The unit sphere is $$ S=\{[1:x_1:x_2:x_3]\mid x_1^2+x_2^2+x_3^2=1\}=\{[x_0:x_1:x_2:x_3]\mid x_1^2+x_2^2+x_3^2=x_0^2\}, $$ while the paraboloid is $$P=\{[1:x_1:x_2:x_3]\mid x_3=x_1^2+x_2^2\}=\{[x_0:x_1:x_2:x_3]\mid x_0x_3=x_1^2+x_2^2,\ x_0\neq 0\}.$$ $P$ has a "point at infinity" $[0:0:0:1]$. What Thurston really means by "paraboloid" when saying "maps the unit sphere to paraboloid" is P with this point at infinity attached, i.e. $\{[x_0:x_1:x_2:x_3]\mid x_0x_3=x_1^2+x_2^2\}$. To find the required map, try something like $[x_0:x_1:x_2:x_3]\mapsto [x_0+x_3:x_1:x_2:x_0-x_3]$.

The point is to understand that all conic hypersurfaces in ℝℙn are projectively equivalent (in other words, in projective geometry there is no distinction among ellipsoid, paraboloid and hyperboloid), and such a hypersurface delimits a convex body which serves as a model of $\mathbb{H}^n$, where geodesics are straight lines.

Your trouble arises because Thurston assumes some familiarity with classical projective geometry. Put n=3 for example. Consider $\mathbb{R}^3\subset\mathbb{RP}^3$ as an affine chart, say, the chart $\{[1:x_1:x_2:x_3]\mid x_i\in\mathbb{R}\}$. The unit sphere is $$ S=\{[1:x_1:x_2:x_3]\mid x_1^2+x_2^2+x_3^2=1\}=\{[x_0:x_1:x_2:x_3]\mid x_1^2+x_2^2+x_3^2=x_0^2\}, $$ while the paraboloid is $$P=\{[1:x_1:x_2:x_3]\mid x_3=x_1^2+x_2^2\}=\{[x_0:x_1:x_2:x_3]\mid x_0x_3=x_1^2+x_2^2,\ x_0\neq 0\}.$$ $P$ has a "point at infinity" $[1:0:0:0]$. What Thurston really means by "paraboloid" when saying "maps the unit sphere to paraboloid" is P with this point at infinity attached, i.e. $\{[x_0:x_1:x_2:x_3]\mid x_0x_3=x_1^2+x_2^2\}$. To find the required map, try something like $[x_0:x_1:x_2:x_3]\mapsto [y_0:y_1:y_2:y_3]$, $y_0=x_0+x_3$, $y_3=x_0-x_3$.

The point is to understand that all conic hypersurfaces in ℝℙn are projectively equivalent (in other words, in projective geometry there is no distinction among ellipsoid, paraboloid and hyperboloid), and such a hypersurface delimits a convex body which serves as a model of $\mathbb{H}^n$, where geodesics are straight lines.

Your trouble arises because Thurston assumes some familiarity with classical projective geometry. Put n=3 for example. Consider $\mathbb{R}^3\subset\mathbb{RP}^3$ as an affine chart, say, the chart $\{[1:x_1:x_2:x_3]\mid x_i\in\mathbb{R}\}$. The unit sphere is $$ S=\{[1:x_1:x_2:x_3]\mid x_1^2+x_2^2+x_3^2=1\}=\{[x_0:x_1:x_2:x_3]\mid x_1^2+x_2^2+x_3^2=x_0^2\}, $$ while the paraboloid is $$P=\{[1:x_1:x_2:x_3]\mid x_3=x_1^2+x_2^2\}=\{[x_0:x_1:x_2:x_3]\mid x_0x_3=x_1^2+x_2^2,\ x_0\neq 0\}.$$ $P$ has a "point at infinity" $[1:0:0:0]$. What Thurston really means by "paraboloid" when saying "maps the unit sphere to paraboloid" is P with this point at infinity attached, i.e. $\{[x_0:x_1:x_2:x_3]\mid x_0x_3=x_1^2+x_2^2\}$. To find the required map, try something like $[x_0:x_1:x_2:x_3]\mapsto [x_0+x_3:x_1:x_2:x_0-x_3]$.

The point is to understand that all conic hypersurfaces in ℝℙn are projectively equivalent (in other words, in projective geometry there is no distinction among ellipsoid, paraboloid and hyperboloid), and such a hypersurface delimits a convex body which serves as a model of $\mathbb{H}^n$, where geodesics are straight lines.

Your trouble arises because Thurston assumes some familiarity with classical projective geometry. Put n=3 for example. Consider $\mathbb{R}^3\subset\mathbb{RP}^3$ as an affine chart, say, the chart $\{[1:x_1:x_2:x_3]\mid x_i\in\mathbb{R}\}$. The unit sphere is $$ S=\{[1:x_1:x_2:x_3]\mid x_1^2+x_2^2+x_3^2=1\}=\{[x_0:x_1:x_2:x_3]\mid x_1^2+x_2^2+x_3^2=x_0^2\}, $$ while the paraboloid is $$P=\{[1:x_1:x_2:x_3]\mid x_3=x_1^2+x_2^2\}=\{[x_0:x_1:x_2:x_3]\mid x_0x_3=x_1^2+x_2^2,\ x_0\neq 0\}.$$ $P$ has a "point at infinity" $[1:0:0:0]$. What Thurston really means by "paraboloid" when saying "maps the unit sphere to paraboloid" is P with this point at infinity attached, i.e. $\{[x_0:x_1:x_2:x_3]\mid x_0x_3=x_1^2+x_2^2\}$.

The point is to understand that all conic hypersurfaces in ℝℙn are projectively equivalent (in other words, in projective geometry there is no distinction among ellipsoid, paraboloid and hyperboloid), and such a hypersurface delimits a convex body which serves as a model of $\mathbb{H}^n$, where geodesics are straight lines.

Your trouble arises because Thurston assumes some familiarity with classical projective geometry. Put n=3 for example. Consider $\mathbb{R}^3\subset\mathbb{RP}^3$ as an affine chart, say, the chart $\{[1:x_1:x_2:x_3]\mid x_i\in\mathbb{R}\}$. The unit sphere is $$ S=\{[1:x_1:x_2:x_3]\mid x_1^2+x_2^2+x_3^2=1\}=\{[x_0:x_1:x_2:x_3]\mid x_1^2+x_2^2+x_3^2=x_0^2\}, $$ while the paraboloid is $$P=\{[1:x_1:x_2:x_3]\mid x_3=x_1^2+x_2^2\}=\{[x_0:x_1:x_2:x_3]\mid x_0x_3=x_1^2+x_2^2,\ x_0\neq 0\}.$$ $P$ has a "point at infinity" $[1:0:0:0]$. What Thurston really means by "paraboloid" when saying "maps the unit sphere to paraboloid" is P with this point at infinity attached, i.e. $\{[x_0:x_1:x_2:x_3]\mid x_0x_3=x_1^2+x_2^2\}$. To find the required map, try something like $[x_0:x_1:x_2:x_3]\mapsto [y_0:y_1:y_2:y_3]$, $y_0=x_0+x_3$, $y_3=x_0-x_3$.

The point is to understand that all conic hypersurfaces in ℝℙn are projectively equivalent (in other words, in projective geometry there is no distinction among ellipsoid, paraboloid and hyperboloid), and such a hypersurface delimits a convex body which serves as a model of $\mathbb{H}^n$, where geodesics are straight lines.