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Use $x>0$ as the parameter of the smooth curve:Its velocity vector and speed areThe unit tangent vector is therefore

The parameter-independent formula for the curvature of a space curve givesDifferentiating,Hence $\kappa$ increases for $0<x<1/\sqrt{2}$ and decreases for $x>1/\sqrt{2}$. Its global maximum occurs at $x=1/\sqrt{2}$ and equalsThe corresponding point is

The tangent vectors ofareHence the first fundamental form has coefficientsPutThe upward unit normal is $N=(-f_u,-f_v,1)/W$, so the second fundamental form has coefficientsThus the two forms are

The Gaussian curvature isSince $EG-F^2=W^2$, the graph formula isas in Gaussian curvature of a graph surface.

For the final claim, fix a point of $\gamma$ and make a rigid motion taking $P$ to the plane $z=0$. The common tangent plane is horizontal, so locally $\Sigma$ is the graph $z=f(u,v)$. Along the projected curve $c(s)$ of tangency,Differentiating the second identity givesBecause $c$ is a smooth curve, $c^{\prime }(s)\ne 0$, so the Hessian is singular. Its determinant is zero, and the graph formula gives $K=0$ at every point of $\gamma$. This is tangency to a plane along a curve forces zero Gaussian curvature.

In the Poincare half-plane model, the Riemannian metricassigns to a smooth curve $\gamma (t)=(x(t),y(t))$ the lengthThe metric is conformal to the Euclidean metric, so hyperbolic and Euclidean angles agree. Its area element is

The two geodesics from $e^{i\alpha }$ and $e^{i\beta }$ to infinity are the vertical lines $x=\cos \alpha$ and $x=\cos \beta$, while the third side is the unit semicircle. The triangle lies above that semicircle, so its area isIts interior angles are $\alpha$, $\pi -\beta$, and zero at the ideal vertex, and henceTriangulating a geodesic polygon with $n$ sides into $n-2$ triangles givesthe area of a hyperbolic geodesic polygon and the polygonal Gauss-Bonnet theorem.

For a closed piecewise smooth curve $\gamma$ avoiding $w$, its winding number about $w$ is

The argument principle says that if a positively oriented closed curve $\gamma$ bounds a domain $U$, and a meromorphic function $F$ has no zeros or poles on $\gamma$, thenwhere zeros and poles are counted with multiplicity.

Suppose $F$ and $G$ are holomorphic on a neighbourhood of $\overline{U}$ andFor $0\le t\le 1$, $F+tG$ has no boundary zero, because a zero would imply $|F|=t|G|<|F|$. Its argument-principle count is integer-valued and continuous in $t$, hence constant. Thus Rouché's theorem states that

Now let $w_0=f(z_0)$. Since $|w_0|<1\le |f|$ on the unit circle, Rouché's theorem shows that $f$ and $f-w_0$ have the same number of zeros in the unit disc. The latter has the zero $z_0$, so $f$ has at least one zero there. For any $w$ with $|w|<1$, the same boundary inequality shows that $f-w$ has the same positive number of zeros as $f$. ThereforeThis is the unit-disc image from a boundary modulus lower bound.

Finally, take a sufficiently small positively oriented circle $C$ around zero. The residueis the winding number of $g(C)$ around zero, so $k\in \mathbb{Z}$. The pole is simple, hence its residue is nonzero and $k\ne 0$. Forwe haveThe second term cancels the complete principal part at zero, so the integer residue of a logarithmic derivative proves that $h^{\prime }/h$ has a removable singularity there.

For $V=Z(f_1,\dots ,f_r)$, $T_{V,p}$ is the common kernel of $d{f}_i(p)$; $\dim V$ is the transcendence degree of its function field, and $p$ is smooth when $\dim T_{V,p}=\dim V$. A nonzero maximal Jacobian minor defines a nonempty open smooth locus (generic rank gives nonemptiness).

A line $ux+vy+wz=0$ corresponds to $[u:v:w]\in ({\mathbb{P}}^2{)}^{\ast }$. Lines through $P=[a:b:c]$ satisfy $au+bv+cw=0$, a projective line. For a smooth curve $F=0$, the dual morphism is $P\mapsto [F_x(P):F_y(P):F_z(P)]$.

For $F=x^3+y^3+z^3$, eliminating $x,y,z$ from $[u:v:w]=[x^2:y^2:z^2]$ givesThe dual sextic is singular: its nine cusps are the tangent lines at the nine flexes of the cubic. Thus points of $X$ mapping to singularities are precisely inflection points, where the tangent has higher contact.

The residual-versus-fitted smooth curve bends downward, indicating nonlinearity, and the spread increases at high fitted values. The Q-Q plot is close centrally but departs in both tails, with observations 12, 17, and 67 notable. The scale-location trend confirms heteroscedasticity. The leverage plot identifies observations 1 and 4 as high leverage and 67 as a large residual, with potentially material Cook distance. The additive Gaussian linear model is therefore doubtful. Inspect those observations, transform sale price (often logarithmically), add nonlinear or interaction terms and omitted predictors such as lot size and age, and use weighted or robust regression if unequal variance remains.

A smooth parametrized curve is regular when its velocity never vanishes.