# Quick Volume

1. Nov 27, 2006

### 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 Rt 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 Rt in words: Rt is the "union of all closed triangular regions defined by vertices fj,fk,fm for all combinations of j,k,m at any $t \in \left[ {a,b} \right]$."

Last edited: Nov 27, 2006
2. Dec 25, 2006

### 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 Rt:
$$V_{net} = \frac{1}{2}\int\limits_a^b {\left\| {\left( {f_1 \left( t \right) - f_2 \left( t \right) } \right) \times \left( {f_1 \left( t \right) - f_3 \left( t \right) } \right)} \right\|dt}$$
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 Rt 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 Rt?
(That is, is there a general formula by which one can calculate this volume?)

Last edited: Dec 25, 2006