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The first diagram simplifies in that the two red edges cancel each other. The resulting gluing diagram is a standard diagram for the projective plane, namely a disk glued to a single circle by a double covering map of the boudary of the disc around the other circle.

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To see that the first space is homeomorphic to $\Bbb R \mathrm P^2$, cut it along the top-left to bottom-right diagonal and glue together the blue edges. You will end up with the standard representation of $\Bbb R\mathrm P^2$ in terms of a square with face identifications. Your second space is a disk (2-cell) glued to a bouquet of two circles $a,b$ (after ...

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First we parameterize our surface by two families of curves $\{u=c\}$ and $\{v=d\}$: The we can see that $$\mathbf r_u = \frac{\partial \mathbf r(u_0,v_0)}{\partial u} = \lim_{h\to 0} \frac{\mathbf r(u_0+h,v_0)-\mathbf r(u_0,v_0)}{h}$$ will be tangent to the curve $u=u_0$ at the point $(u_0,v_0)$ and thus also tangent to the surface. Likewise for $\... 2 This is actually one of the standard formulas in a first-year calculus course for the surface area of a surface of revolution. You want to compute the integral $$\int_0^{2\pi}\int_0^r \phi(u)\sqrt{1+\phi'(u)^2}\,du\,d\theta.$$ (This is the surface obtained by rotating the graph$y=\phi(x)$about the$x$-axis.) Alternatively, you can derive this using the ... 1 Since you say you were trying to glue a handkerchief, I suppose you would like some geometric intuition on how these gluing diagrams are interpreted. The study of orbifolds (quotients of manifolds by symmetry groups satisfying certain constraints; any discrete group will do) provides a generalised, natural way to understand and notate these spaces. I will ... 1 Let's call$f_0(y)$the values on the segment$\{0\}\times[0,1]$,$f_1(y)$the values on$\{1\}\times [0,1]$, and similarly$g_0(x)$and$g_1(x)$. A continuous function interpolating them is $$\frac{[(1-x)f_0(y) + xf_1(y)]y(1-y) + [(1-y)g_0(x) + yg_1(x)]x(1-x)}{x(1-x) + y(1-y)}$$ 1 Yes,$X \setminus U$must be compact. To prove this, first let's enlarge the set$K$somewhat: we may include$K$into a subset$K \subset Y$such that$Y$is a smooth, compact, surface-with-boundary. This is not too hard to see: take a collection of smooth balls whose interiors cover$K$; by compactness finitely many of these balls suffices; we may perturb ... 1 So after some time I got both of the algorithms working. In scenes with surfaces of lower degree, algorithm by Mann and DeRose seems to be faster, in other scenes (about 80% of scenes I test my ray tracer with) the algorithm by Sederberg is faster and requires a constant amount of memory so it is more suitable for GPU (no dynamic allocation in OpenCL 1.2). ... 1 No no no! Everyone is talking about the perpendicular vector to the surface. What does this have to do with surface area. If you have two perpendicular vectors$a\frac{d}{dx}$and$b\frac{d}{dy}$on the plane, then they make a rectangle of area$ab$. What if you have two vectors$ a_1\frac{d}{dx}+b_1\frac{d}{dy}$and$a_2\frac{d}{dx}+b_2\frac{d}{dy}$, then ... 1$\newcommand{\dd}{\partial}$No claim of elegance, but Cartesian coordinates handle both questions, and the answers are "yes": Up to translation, a general ellipsoid can be written in the form $$Ax^{2} + By^{2} + Cz^{2} + 2(Dxy + Exz + Fyz) = 1 \tag{1}$$ for some positive-definite coefficient matrix$$\left[\begin{array}{@{}ccc@{}} A & D & E \\ D &... 1 After perusing your Wikipedia link, "I don't know for sure", but here's the explanation that seems most likely to me (a geometer who knows next to nothing about control theory). The state of a system under sliding mode control is modeled as a point in some phase space, a mathematical object encoding both physical configuration (position) and infinitesimal ... 1 If you can put your equation in to this form$\mathbf x^T A \mathbf x = K$and since A is a symmetric matrix, you can diagonalize it, and not only diagonalize it, diagonalize it with ortho-normal P.$\mathbf x^T P^T D P\mathbf x = K\\ (P\mathbf x)^T D P\mathbf x = K$The matrix$P\$ represents a change to the coordinate system -- a basis change. And since ...

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To address the question as clarified in the comments: Generally, you're asking questions of differential topology. The book of that title by Victor Guillemin and Alan Pollack is a clear, detailed introduction to the ideas sketched here. Morse Theory by John Milnor may also be of interest. tl; dr: The following tools are useful for studying level sets of ...

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