3. Representing constraints (40 points) Constraints represent restrictions, which can be modeled mathematically as boundaries. One...

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3. Representing constraints (40 points) Constraints represent restrictions, which can be modeled mathematically as boundaries. One of the simplest and yet powerful ways to represent constraints is through linear equality and inequality equations. However, the same constraint can be *parameterized* in many ways and the choice of this parameterization has very important implications on system function. In this problem I will ask you to try to understand the simplest case, which is itself complicated - representing a *triangle* as an equation. To define terms, let's characterize an equation for a triangle in terms of a locus of points $x,y$ where a function $f_{triangle}(x,y)$ goes to zero. This is like defining a circle with radius $r$ and center $x_c.y_c$ as $f_{circle}(x,y) = (x-x_c)^2+(y-y_c)^2-r^2$. The idea is that we should take some parametrized description of the triangle (edges, vertices, angles) as input and construct a function that goes to zero only on the triangle. (10 points each) (a) Vertex form: Show that we can describe the interior of the triangle as a weighted combination of the vertices where the weights sum to one and are all positive. Find conditions on the weights that characterize the triangle (the boundary). (b) Line form: Describe the triangle using three linear equations for the boundary lines in normal form (e.g. $a x + by +c*1 = 0$) and show that the triangle equation can be expressed as the product of the three equations. (c) Normal form: Describe the edges of the triangle using the (dual) normal vectors to the sides of the triangle, where the normal vectors start at the triangle's centroid. Show that the equation of the line is given by the zero locus of the minimum of the dot product of the normals with points in the plane described as vectors with origin at the centroid. (d) Exponential form: Using form c, show that an equation of the can be formed by the zero locus of $1-\sum_i \exp(z_i(x))$ where $z_i(x)$ is the value of the dot product of a point with side $i$. 3. Representing constraints (40 points) Constraints represent restrictions, which can be modeled mathematically as boundaries. One of the simplest and yet powerful ways to represent constraints is through linear equality and inequality equations. However, the same constraint can be *parameterized* in many ways and the choice of this parameterization has very important implications on system function. In this problem I will ask you to try to understand the simplest case, which is itself complicated - representing a *triangle* as an equation. To define terms, let's characterize an equation for a triangle in terms of a locus of points $x,y$ where a function $f_{triangle}(x,y)$ goes to zero. This is like defining a circle with radius $r$ and center $x_c.y_c$ as $f_{circle}(x,y) = (x-x_c)^2+(y-y_c)^2-r^2$. The idea is that we should take some parametrized description of the triangle (edges, vertices, angles) as input and construct a function that goes to zero only on the triangle. (10 points each) (a) Vertex form: Show that we can describe the interior of the triangle as a weighted combination of the vertices where the weights sum to one and are all positive. Find conditions on the weights that characterize the triangle (the boundary). (b) Line form: Describe the triangle using three linear equations for the boundary lines in normal form (e.g. $a x + by +c*1 = 0$) and show that the triangle equation can be expressed as the product of the three equations. (c) Normal form: Describe the edges of the triangle using the (dual) normal vectors to the sides of the triangle, where the normal vectors start at the triangle's centroid. Show that the equation of the line is given by the zero locus of the minimum of the dot product of the normals with points in the plane described as vectors with origin at the centroid. (d) Exponential form: Using form c, show that an equation of the can be formed by the zero locus of $1-\sum_i \exp(z_i(x))$ where $z_i(x)$ is the value of the dot product of a point with side $i$.