One parameter family of functions describing the curve

[boyboy400]

"one parameter family of functions" describing the curve

Homework Statement

Let a cube of unit length be represented by the set of vectors such that three of the cube's sides are aligned with the three orthonormal basis set vectors e1 e2 and e3, and one of the cube vertices lies on the origin of the coordinate system. Let Ω be the deformed configuration such that f:X belonging to Ω0 -> x=f(X) belonging to Ω is smooth, differentiable and bijective defined as:
(X1,X2 and X3 are three original positions)

x1=1.1X1+0.02X1^2+0.01X2+0.03X3
x2=0.001X1+0.9X2+0.003X3
x3=0.001X1+0.005X2+0.009X3^2+0.9X3

a- The displacement function u
b- The new position of the 8 vertices of the cube
c- The deformed curves of three sides of the cube that are aligned with the basis vectors e1, e2 and e3.

Homework Equations

The Attempt at a Solution

a- It's super easy: u=x-X so we already have both and u can be obtained as a function of X1 and X2 and X3

b- Well we already have the original positions of the vertices (X) so by substituting them into the x equations the new positions are obtained.

c- I don't understand it. I know that the answer wants the description of the curve created after the edge deforms. The instructor wants the "one parameter family of functions" describing the curve.
But I don't understand what this "one parameter family of functions" means! Can anyone please help me? I'm in urgent need of help as usual :(
But I don't understand what it means

Even if you have any sort of understanding or interpretation please please let me know. Thanks

 

[mighty2000]

in advance!
Thank you for your question. The "one parameter family of functions" in this context refers to a set of functions that are all related to each other through a single parameter. In this case, the parameter is X1, X2, and X3, which are the original positions of the cube's vertices.

To understand this better, let's take a closer look at the displacement function u. As you correctly stated, u can be obtained as a function of X1, X2, and X3. This means that u is a function that takes in the original positions of the cube's vertices and outputs the displacement of those vertices. So, for each set of X1, X2, and X3, we get a different displacement function.

Now, let's consider the three sides of the cube that are aligned with the basis vectors e1, e2, and e3. Each of these sides can be described by a curve in 3-dimensional space. This curve can be represented by a function of X1, X2, and X3. However, since the cube is deformed, these curves will also be affected by the displacement function u. This means that for each set of X1, X2, and X3, we get a different curve.

So, the "one parameter family of functions" in this case refers to the set of functions that describe the curves of the three sides of the cube, with each function being related to a different set of X1, X2, and X3. I hope this helps to clarify the concept for you.