Plotting a metric space a>0, c>0; 4ac-b2>0

[fabbi007]

Metric given by (d) in the textbook on page 48 in the following url. How could we plot this function and characterize what the metric looks like for varying a, b, c. Start by plotting the case when d(x,0)=1, a=b=c=1. Vary a, b, and c individually. Show plots for each case and Characterize how the shape of the metric changes.


http://books.google.com/books?id=t3...A48#v=onepage&q=4ac-b2>0 metric space&f=false


My attempt:

The equation yields either a parabola or ellipse. Not sure where to start. A direction or hint would help. Thanks for your responses.

 

[jvicens]

To plot this function, we can use a graphing software such as Desmos or Wolfram Alpha. We can also use a spreadsheet program like Excel to create a table of values and then plot them on a graph.

Let's start by considering the case when d(x,0)=1, a=b=c=1. This means that the equation becomes:

d(x,y) = √(x^2 + y^2) + √((x-1)^2 + y^2) + √(x^2 + (y-1)^2) = 1

This can be simplified to:

√(2x^2 + 2y^2 - 2x + 2y + 2) = 1

Squaring both sides, we get:

2x^2 + 2y^2 - 2x + 2y + 2 = 1

Rearranging, we get the equation of an ellipse:

x^2 + y^2 - x + y + 1 = 0

Now, let's vary a, b, and c individually. We will keep the other variables constant at 1.

When we vary a, the coefficient of x^2 changes and therefore, the shape of the ellipse changes. If a is decreased, the ellipse will become wider and flatter, while if a is increased, the ellipse will become narrower and taller.

When we vary b, the coefficient of y^2 changes and therefore, the shape of the ellipse changes. If b is decreased, the ellipse will become wider and flatter, while if b is increased, the ellipse will become narrower and taller.

When we vary c, the coefficients of x and y change and therefore, the position of the ellipse changes. If c is decreased, the center of the ellipse will shift to the left and down, while if c is increased, the center of the ellipse will shift to the right and up.

We can plot these functions and see how the shape of the metric changes for different values of a, b, and c. For example, when a=2, b=1, and c=1, the equation becomes:

2x^2 + y^2 - 2x + y + 1 = 0

This results in a narrower and taller ellipse with a center shifted to the left and down compared to the original ellipse when a=b=c=1