Curvilinear squares method

These dimensions are illustrated in Fig. 3-4. Note that the shape factor per unit depth is given by the ratio MIN when the curvilinear-squares method is used for calculations. The use of the shape factor for calculation purposes is illustrated in Examples 3-1 and 3-2. 

Figure 4.4-1. Graphical curvilinear-square method for two-dimensional heat conduction in a rectangular flue.

The accuracy of this method is dependent entirely on the skill of the person sketching the curvilinear squares. Even a crude sketch, however, can frequently help to give fairly good estimates of the temperatures that will occur in a body and these estimates may then be refined with numerical techniques discussed in Sec. 3-5. An electrical analogy may be employed to sketch the curvilinear squares, as discussed in Sec. 3-9. 

Once the electric potential is impressed on the paper, an ordinary voltmeter may be used to plot lines of constant electric potential. With these constant-potential lines available, the flux lines may be easily constructed since they are orthogonal to the potential lines. These equipotential and flux lines have precisely the same arrangement as the isotherms and heat-flux lines in the corresponding heat-conduction problem. The shape factor is calculated immediately using the method which was applied to the curvilinear squares. 

Curvilinear-Squares Graphical Method. Repeat Example 4.4-1 but with the... 

Curvilinear regression should not be confused with the nonlinear regression methods used to estimate model parameters expressed in a nonlinear form. For example, the model parameters a and b in y = axb cannot be estimated by a linear least-squares algorithm. Information in Chapter 7 describes nonlinear approaches to use in this case. Alternatively, a transformation to a linear model can sometimes be used. Implementing a logarithmic transformation on yt = ax/ produces the model log y = log a + blog x which can now be utilized with a linear least-squares algorithm. The literature [4, 5] should be consulted for additional information on linear transformations. 

More standard points are needed to adequately define a curvilinear function than to define a linear function. Weighted least-square regression methods are... 

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