Finding Volume Traversed by Rt from a to b

[bomba923]

(Wow...it's been over three months since I posted anything...)
Anyhow,

Given continuous functions
$$\begin{gathered} f_1 :\left[ {a,b} \right] \to \mathbb{R}^3 \hfill \\ \vdots \hfill \\ f_n :\left[ {a,b} \right] \to \mathbb{R}^3 \hfill \\ \end{gathered}$$
for which
$$\exists g:\left[ {a,b} \right] \to \mathbb{R}^3 {\text{ such that }}f_1 \cdot g = \cdots = f_n \cdot g$$

define $\forall t \in \left[ {a,b} \right]$
$$R_t = \left\{ {\left( {x,y,z} \right)\left| {\left( {x,y,z} \right) \cdot g = f_1 \cdot g} \right.} \right\} \cap$$
$$\bigcup\limits_{\begin{subarray}{l} j < k < m \leqslant n, \\ \left( {j,k,m} \right) \in \mathbb{N}^3 \end{subarray}} {\left\{ {\left( {x,y,z} \right)\left| \begin{gathered} \left( {y - y_j } \right)\left( {x_j - x_k } \right)\left| {\begin{array}{*{20}c} {x_j - x_k } & {y_j - y_k } \\ {x_j - x_m } & {y_j - y_m } \\ \end{array} } \right| \leqslant \left( {x - x_j } \right)\left( {y_j - y_k } \right)\left| {\begin{array}{*{20}c} {x_j - x_k } & {y_j - y_k } \\ {x_j - x_m } & {y_j - y_m } \\ \end{array} } \right| \wedge \hfill \\ \left( {y - y_j } \right)\left( {x_j - x_m } \right)\left| {\begin{array}{*{20}c} {x_j - x_m } & {y_j - y_m } \\ {x_j - x_k } & {y_j - y_k } \\ \end{array} } \right| \leqslant \left( {x - x_j } \right)\left( {y_j - y_m } \right)\left| {\begin{array}{*{20}c} {x_j - x_m } & {y_j - y_m } \\ {x_j - x_k } & {y_j - y_k } \\ \end{array} } \right| \wedge \hfill \\ \left( {y - y_k } \right)\left( {x_k - x_m } \right)\left| {\begin{array}{*{20}c} {x_k - x_m } & {y_k - y_m } \\ {x_k - x_j } & {y_k - y_j } \\ \end{array} } \right| \leqslant \left( {x - x_k } \right)\left( {y_k - y_m } \right)\left| {\begin{array}{*{20}c} {x_k - x_m } & {y_k - y_m } \\ {x_k - x_j } & {y_k - y_j } \\ \end{array} } \right| \wedge \hfill \\ \left( {z - z_j } \right)\left( {x_j - x_k } \right)\left| {\begin{array}{*{20}c} {x_j - x_k } & {z_j - z_k } \\ {x_j - x_m } & {z_j - z_m } \\ \end{array} } \right| \leqslant \left( {x - x_j } \right)\left( {z_j - z_k } \right)\left| {\begin{array}{*{20}c} {x_j - x_k } & {z_j - z_k } \\ {x_j - x_m } & {z_j - z_m } \\ \end{array} } \right| \wedge \hfill \\ \left( {z - z_j } \right)\left( {x_j - x_m } \right)\left| {\begin{array}{*{20}c} {x_j - x_m } & {z_j - z_m } \\ {x_j - x_k } & {z_j - z_k } \\ \end{array} } \right| \leqslant \left( {x - x_j } \right)\left( {z_j - z_m } \right)\left| {\begin{array}{*{20}c} {x_j - x_m } & {z_j - z_m } \\ {x_j - x_k } & {z_j - z_k } \\ \end{array} } \right| \wedge \hfill \\ \left( {z - z_k } \right)\left( {x_k - x_m } \right)\left| {\begin{array}{*{20}c} {x_k - x_m } & {z_k - z_m } \\ {x_k - x_j } & {z_k - z_j } \\ \end{array} } \right| \leqslant \left( {x - x_k } \right)\left( {z_k - z_m } \right)\left| {\begin{array}{*{20}c} {x_k - x_m } & {z_k - z_m } \\ {x_k - x_j } & {z_k - z_j } \\ \end{array} } \right| \wedge \hfill \\ \left( {z - z_j } \right)\left( {y_j - y_k } \right)\left| {\begin{array}{*{20}c} {y_j - y_k } & {z_j - z_k } \\ {y_j - y_m } & {z_j - z_m } \\ \end{array} } \right| \leqslant \left( {y - y_j } \right)\left( {z_j - z_k } \right)\left| {\begin{array}{*{20}c} {x_j - x_k } & {z_j - z_k } \\ {x_j - x_m } & {z_j - z_m } \\ \end{array} } \right| \wedge \hfill \\ \left( {z - z_j } \right)\left( {y_j - y_m } \right)\left| {\begin{array}{*{20}c} {y_j - y_m } & {z_j - z_m } \\ {y_j - y_k } & {z_j - z_k } \\ \end{array} } \right| \leqslant \left( {y - y_j } \right)\left( {z_j - z_m } \right)\left| {\begin{array}{*{20}c} {x_j - x_m } & {z_j - z_m } \\ {x_j - x_k } & {z_j - z_k } \\ \end{array} } \right| \wedge \hfill \\ \left( {z - z_k } \right)\left( {y_k - y_m } \right)\left| {\begin{array}{*{20}c} {y_k - y_m } & {z_k - z_m } \\ {y_k - y_j } & {z_k - z_j } \\ \end{array} } \right| \leqslant \left( {y - y_k } \right)\left( {z_k - z_m } \right)\left| {\begin{array}{*{20}c} {y_k - y_m } & {z_k - z_m } \\ {y_k - y_j } & {z_k - z_j } \\ \end{array} } \right| \hfill \\ \end{gathered} \right.} \right\}}$$
where $$\forall i > 0,\;\left( {x_i ,y_i ,z_i } \right) = f_i \left( t \right)$$

Find the net volume traversed by R_t from $t=a$ to $t=b$ if
$$\exists p,q \in \left( {a,b} \right):\left( {R_p - \partial R_p } \right) \cap \left( {R_q - \partial R_q } \right) \ne \emptyset \, \wedge \, p \ne q$$

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*Edit: it may be simpler to describe R_t in words: R_t is the "union of all closed triangular regions defined by vertices f_j,f_k,f_m for all combinations of j,k,m at any $t \in \left[ {a,b} \right]$."

 

[bomba923]

Anybody?

Well, let's take the simplest case (n=3), where we have three continuous functions
$$\begin{gathered} f_1 :\left[ {a,b} \right] \to \mathbb{R}^3 \hfill \\ f_2 :\left[ {a,b} \right] \to \mathbb{R}^3 \hfill \\ f_3 :\left[ {a,b} \right] \to \mathbb{R}^3 \hfill \\ \end{gathered}$$
and
$$\forall t \in \left[ {a,b} \right]$$, let $$R_t$$ be the open triangular region with vertices f₁(t), f₂(t), and f₃(t).

To simplify matters, assume that $\forall p,q \in \left( {a,b} \right),\;\left( {R_p \cap R_q \ne \emptyset } \right) \to \left( {p = q} \right)$.
Find the net volume traversed by $$R_t$$ from t=a to t=b
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Initially, one might guess the net volume to simply the sum of the areas of R_t:
$$V_{net} = \frac{1}{2}\int\limits_a^b {\left\| {\left( {f_1 \left( t \right) - f_2 \left( t \right) } \right) \times \left( {f_1 \left( t \right) - f_3 \left( t \right) } \right)} \right\|dt}$$
But, that is false!
Consider
$$\begin{gathered} f_1 \left( t \right) = \left( {t,0,0} \right) \hfill \\ f_2 \left( t \right) = \left( {0,t,0} \right) \hfill \\ f_3 \left( t \right) = \left( {0,0,t} \right) \hfill \\ \end{gathered}$$
for
$$0 \leqslant t \leqslant 1$$

The region traversed by R_t is the tetrahedron with vertices (0,0,0), (1,0,0), (0,1,0), (0,0,1).
Its volume is simply
$$\int\limits_0^1 {\int\limits_0^{1 - x} {\left( {1 - x - y} \right)dy} dx} = \frac{1}{6}$$
However,
$$\frac{1}{2}\int\limits_0^1 {\left\| {\left( {f_1 \left( t \right) - f_2 \left( t \right)} \right) \times \left( {f_1 \left( t \right) - f_3 \left( t \right)} \right)} \right\|dt} = \frac{{\sqrt 3 }}{6} \ne \frac{1}{6}$$
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So, given three continuous functions
$$\begin{gathered} f_1 :\left[ {a,b} \right] \subset \mathbb{R} \to \mathbb{R}^3 \hfill \\ f_2 :\left[ {a,b} \right] \subset \mathbb{R} \to \mathbb{R}^3 \hfill \\ f_3 :\left[ {a,b} \right] \subset \mathbb{R} \to \mathbb{R}^3 \hfill \\ \end{gathered}$$

How can I calculate the net volume traversed by R_t?
(That is, is there a general formula by which one can calculate this volume?)

 

[tacman]

Wow, that is quite a complex problem! It seems like you are dealing with multiple continuous functions and finding the volume traversed by their corresponding points. It's great to see that you are still actively working on math problems after three months.

From what I understand, Rt is essentially a collection of closed triangular regions defined by the vertices of the functions f1 to fn at any given t in the interval [a, b]. And you are trying to find the volume of this entire collection.

To find the net volume traversed by Rt from t=a to t=b, it seems like you would need to determine the overlapping regions between the different triangular regions. This is where the conditions of p and q being different and the intersection of their boundaries come into play.

Without actually seeing the specific functions and points involved, it's hard for me to give a concrete answer. But it seems like you have set up a mathematical framework to tackle this problem. Keep up the good work and I hope you are able to find a solution to this problem!