By Dawson C.N., Martinez-Canales M.L.
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Extra resources for Acharacteristic-Galerkin Aproximation to a system of Shallow Water Equations
Express the Cartesian point (1, 1, 1) in cylindrical polar coordinates. 46 3. 2. Express the Cartesian point (0, 2, 2) in spherical polar coordinates. 3. What is the equation of a vertical semicircle of radius 1, centred at the origin, and whose projection onto the x, y-plane lies on the Cartesian line y = x in the ﬁrst quadrant? Give your answer in spherical polar coordinates. 4. What is the equation of a horizontal circle of radius 4 and a constant height of 3, whose projection onto the x, y-plane is centred at the origin?
But how about if we take away the constraint of stating the distance from the origin? Then we’re just left with an angle from the positive x-axis – and this describes a line! 2 Lines and Circles 37 Let’s take a look at an example. If we are given the information θ = 3π 4 , but no value of r, then we have an inﬁnite number of points that satisfy the condition. 4. 4 Quite impressive, I’m sure you’ll agree. In Cartesian coordinates, we would have had to have described the line y = −x, and also the restriction that we only want the part of the line that is in the second quadrant.
Thankfully, all we need to do is describe another line that our line is perpendicular to, and then we know what “steepness” we’re going to be dealing with. But why is describing this new line, which our line is perpendicular to, any better than describing our line in the ﬁrst place? Easy: 38 3. Polar Coordinates We can choose our perpendicular line to be a line that goes through the origin, which we already know how to completely describe! If we were trying to convert the line y = x + 6 from Cartesian to polar coordinates, we could see that this line is deﬁnitely perpendicular to the line y = −x, which can be described as θ = 3π 4 for the second quadrant.
Acharacteristic-Galerkin Aproximation to a system of Shallow Water Equations by Dawson C.N., Martinez-Canales M.L.