# Tag Info

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The following inputs will plot the following 6 hearts in the picture below respectively. ContourPlot[(x^2 + y^2 - 1)^3 - x^2 y^3 == 0, {x, -1.5, 1.5}, {y, -1.5, 1.5}, MaxRecursion -> 5] ContourPlot[x^2 + (y - (2 (x^2 + Abs[x] - 6))/(3 (x^2 + Abs[x] + 2)))^2 == 36, {x, -9, 9}, {y, -9, 9}, MaxRecursion -> 5] ContourPlot[x^2 + (5/4 y - Sqrt[Abs[x]])^2 ...

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We need two ingredients, the first one is the total curvature formula $$\int_C k(s)ds= 2\pi$$ for every smooth simple closed planar curve $C$. Here $k(s)$ is the curvature function. If $k(s)\ge K>0$ for all $s$ (I do not like using $L$ for a curvature bound), we obtain $$K L(C)\le \int_C k(s) ds=2\pi,$$ where $L=L(C)$ is the length of $C$. Thus, $$... 2 Sorry but this is not the answer but too long for a comment: Probably the easiest verification is to type the equation on Google you'l be surprised : The easiest way is to Google :2 ... 2 Unless the directions specified by f'(A), f'(B) are in line with the straight line connecting the points I suspect that you can't find such a function. In particular, there seems to be a family of curves connecting the points, and satisfying your conditions but where the length of the family approaches that of the straight line. Of course, this means that ... 2$$ \begin{align*} \frac{1}{h} & \left[ \langle \gamma(t+h),\, \eta(t+h)\rangle - \langle \gamma(t),\, \eta(t) \rangle \right] \\ & = \frac{1}{h} \left[ \langle \gamma(t+h),\, \eta(t+h)\rangle - \langle \gamma(t),\, \eta(t+h)\rangle \right] + \frac{1}{h} \left[ \langle \gamma(t),\, \eta(t+h)\rangle - ...

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First you have 2 given points and the direction the car is facing: Then draw the 2 tangent circles to each position with a radius given by the turn radius of the car: Then find the line segments that are tangent two circles, one from each given point. Only 3 are shown here, but since each circle contributes two locations, there are a total of 16 ...

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If $C$ is a non-singular curve of degree $d$ then the degree of its dual curve $C^*$ is indeed $d^*=d(d-1)$. However this does not hold for singular curves. Plücker formula gives the degree of $C^*$ in terms of the degree and singularities of $C$. The formula is $d^*=d(d-1)-2\delta-3\kappa$, where $\delta$ is the number of ordinary nodes and $\kappa$ is the ...

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I don't think the line is that hard to parametrize that you should give up on that approach. But anyway, here is a different one. Two manifolds are tangent if their tangent spaces coincide (in case of unequal dimensions, are related by inclusion). Equivalently, one can consider the orthogonal complement of the tangent space, which in the present case is ...

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