Developing a Model for Curve-Fitting a Tree Stem’s Cross-Sectional Shape and Sapwood–Heartwood Transition in a Polar Diagram System Using Nonlinear Regression

Abstract

A function from the domain (x-set) to the codomain (y-set) connects each x element to precisely one y element. Since each x-point originating from the domain corresponds to two y-points on the graph of a closed curve (i.e., circle, ellipse, superellipse, or ovoid) in a rectangular (Cartesian) diagram, it does not fulfil the function’s requirements. This non-function phenomenon obstructs the nonlinear regression application for fitting observed data resembling a closed curve; thus, it requires transforming the rectangular coordinate system into a polar coordinate system. This study discusses nonlinear regression to fit the circumference of a tree stem’s cross-section and its sapwood–heartwood transition by transforming rectangular coordinates (x, y) of the observed data points’ positions into polar coordinates (r, θ). Following a polar coordinate model, circular curve fitting fits a log’s cross-sectional shape and sapwood–heartwood transition. Ellipse models result in better goodness of fit than circular ones, while the rotated ellipse is the best-fit one. Deviation from the circular shape indicates environmental effects on vascular cambium differentiation. Foresters have good choices: (1) continuing using the circular model as the simplest one or (2) changing to the rotated ellipse model because it gives the best fit to estimate a tree stem’s cross-sectional shape; therefore, it is more reliable to determine basal area, tree volume, and tree trunk biomass. Computer modelling transforms the best-fit model’s formulas of the rotated ellipse using Python scripts provided by Wolfram engine libraries.

1. Introduction

Nonlinear regression is part of statistical analysis methods to curvilinearly fit the observed data to follow a function of indicating predictor(s). Nonlinear regression takes a y~f(x, β) nonlinear function as the statistical model to attribute the corresponding observed response variable (y) to the indicating predictor (x), and foresters frequently employ it to analyze many natural behaviors’ inter-dependency in the forest and environment. Nonlinear regression is reliable for analyzing a tree’s xylem growth curve (i.e., fiber length, cell-wall thickness, and microfibril angle) [1,2] and determining its juvenile–mature periphery. Construction engineers also rely on nonlinear regression to empirically justify the service environment effects on the structural properties of a wooden building member [3,4], estimating its residual service life [5,6,7], and even determining the time interval of people’s comfort in an environment that is neighbor to forest stands [8]. A material’s strength test’s load–displacement curve resembles a curvilinear shape. Forest engineers apply nonlinear regression to develop an allometric equation in forest inventory to estimate biomass, carbon stocks, and wood production [9,10,11,12,13,14]. However, it is dispiriting that current remote sensing and sensors still insist on advocating an algorithm following the linear correlation between several measured predicting and dependent variables to estimate the aboveground biomass [15]. This study developed alternative calculation methods [16] related to nonlinear regression.

Tree trunk biomass is an essential parameter in estimating forest productivity [17], growth [18], and carbon sequestration [19]. A stem’s cross-sectional shape and size change with the tree’s growth. An accurate description of the dynamic interrelation is necessary for describing a tree’s ability to absorb carbon from the atmosphere and store it in its trunks. Forest science traditionally simplifies a tree’s basal area and log cross-section as a circular shape [20]. It has been widely used in forest inventory, annual cutting allowance determination, the wood market, and carbon sequestration assessments. Under only genetic influence during the tree growth periods, a healthy tree’s cambium cell differentiation to produce xylem proceeds at a similar rate along the tree circumference; thus, the annual-ring shape and tree trunk cross-section should be circular [21]. The deviation from the circular form of a stem’s cross-sectional shape indicates the effect of external factors, such as local climate, environmental disturbance (e.g., light intensity and direction, water availability, wind force, and snow mound mass), biogeophysical aspects (e.g., ground position and trunk slope), competition, and succession during tree growth. Several standard 2D geometric shapes (ellipse, superellipse, or ovoid) may resemble these varied cross-sectional shapes. The peripheral circumference of a log’s cross-section can be digitized, and the data plot forms a curve that a mathematical equation can fit.

Digitization and curve fitting may also be run for each annual tree ring and the sapwood–heartwood transition. Wood is produced by many plants, including gymnosperms and angiosperms, and the different growth rates in each season (i.e., spring–summer–autumn–winter in temperate regions or rainy–drought in the tropical region) tag early and late wood cell gradation. Gradation creates annual tree rings. Dendrochronology reads annual tree-ring variations to analyze a tree’s archaeological and historical structures for reconstructing past climates and significantly contributes to environmental science development [22]. Tree-ring chronologies, which may contain dimensional variations and circular shape deviations, provide high-frequency signals [23] that record a tree’s responses to the silvicultural system [24,25] and its habitat’s environmental condition history (i.e., climate [26,27,28,29,30,31,32,33,34], pollutants [35,36], and water–heat stress [37]) during its growing periods yielding the xylem.

The outer part of the wood that lies between the cambium and the heartwood is called sapwood. Sapwood has living parenchymal cells (including vertically oriented longitudinal parenchymal cells and horizontally oriented ray cells) and is physiologically active in storing foods. About 10% of sapwood cells are alive on average, and they play a role in transporting water and minerals to the tree crown; therefore, they contain more water and are more permeable than heartwood. In addition to cambial age and axial height [38], tree inclination [39] affects the sapwood’s anatomical features. Heartwood formation results in the death of parenchymal cells. When the xylem becomes older, the cells die and produce heartwood, which is no longer involved in the physiological process but continues providing mechanical support. Heartwood tissue generally produces a specific color formation [40] because of the deposits’ enhancement of staining extractive content. Contrast color indicates higher extractive content, significantly improving the wood’s natural durability. The degradation and incrustation of pores in the xylem vascular elements with extractive substances accompany heartwood formation, significantly reduce wood permeability, lead to water transport cessation, and prevent wood-decaying organism penetration [41,42,43]. Scented extractive substance contents in heartwood, namely, terpenoids (i.e., monoterpenoids and sesquiterpenes), are synthesized, accumulated, and shaped for fragrance formation [44]. Although the circumference of the sapwood–heartwood transition periphery does not follow a certain annual tree ring and may form an irregular pattern, some idealization to become a standard geometric section may be possible to be promoted.

This study discusses the theoretical basis and application of nonlinear regression and creates alternative computation tools [16] to estimate a tree stem’s cross-sectional shape and its sapwood–heartwood transition by transforming rectangular coordinates (x, y) of the raw data points’ positions into polar coordinates (r, θ). Some 2D closed curved geometric shapes, such as circles, ellipses, or superellipses, may fit a log’s cross-sectional shape and sapwood–heartwood transition. The mathematical models promoted in this study may become a starting point to develop an algorithm to formulate the best-fit curve of a tree’s cross-sectional shape at each growth stage. Therefore, the signal recorded in the annual tree ring can be read more precisely by measuring its deviation from the standard geometric shape.

2. Mathematical Basis

The function idealizes the dependency of a quantity variation on another quantity. A function from the domain (x-set) to the codomain (y-set) connects each element of x to exactly an element of y. The f, g, and h letters may represent functions, and f(x), g(x), and h(x) represent the function value of the x-element. The set of all pairs of (x, f(x)) uniquely describes a function. Each point has a pair of coordinates, and all points are plotted in a rectangular (Cartesian) diagram to form a graph of the function. A rectangular diagram plots a point position as a linear distance from the origin about two or three mutually perpendicular axes. The origin is the center of the coordinates where the axes intersect {(0, 0) or (0, 0, 0)}, and a point plotted in the rectangular diagram is defined as the coordinate {(x, y) or (x, y, z)}. Each set of coordinates uniquely establishes the position of a point in a 2D plane or 3D space. Rectangular diagrams are well-known and used in many applications to graph a function.

Several functions (including interpolation of circular, ellipse, superellipse, or ovoid) may be more effectively expressed in nonlinear coordinate systems such as a polar than in a rectangular diagram. In a rectangular diagram, each x-point originating from the domain corresponds to two y-points on the graph of a circle, ellipse, superellipse, or ovoid; thus, they are not eligible for a function’s requirements. This non-function phenomenon prohibits the nonlinear regression application for fitting the observed data; therefore, this study suggests transforming the rectangular coordinate system into a polar coordinate system to solve these specific cases. The curve fitting of the circumference of a log and the leaves’ shape require nonlinear regression analysis methods at polar diagrams that consider the transformation of rectangular coordinate (x, y) position points into polar coordinates (r, θ). In polar coordinates, a point position is represented by a straight distance from the origin (0, 0) and an angle to the positive horizontal x-axis. The straight distance from the origin is the radius (r), and the counterclockwise rotation angle centered from the origin is θ.

In contrast to rectangular diagrams, the polar coordinate system allows an infinite number of sets of coordinates to describe any point (for example, the locations of points (r, θ), (r, θ + 2π), and (r, θ + 4π) are the same). The angle value (θ) is positive (+) to find the coordinates of the angle that rotated in the counterclockwise direction and a negative angle value (−) for the clockwise one. The radial coordinate (r) can also be positive or negative. Negative radial coordinates (−r) indicate that the angular coordinates are in the opposite quadrant of the destination point. A negative radius moves the point back to the destination point. Data transformation from the (x, y) rectangular coordinate system into the (r, θ) polar coordinate system converts the non-function form of a circle, ellipse, superellipse, or ovoid; then, may result in a function that could be well-fitted using nonlinear regression.

2.1. Circle

Equation (1) represents the standard formula of a circular shape with center (p, q) and radius r = a in a Cartesian (rectangular) coordinate system. Equation (1) can be rearranged to become Equation (2), then transformed into Equation (3) to build the polar form. In its polar form, the circle curve is drawn from the r = f(θ) function for 0 < θ < 2π, where the r cos θ, r sin θ, and r² substitute x, y, and x²+y², respectively. Some algebra steps rearrange Equation (3) to become Equation (4), which one can solve for r to become Equation (5):

$${\left(x-p\right)}^2+{\left(y-q\right)}^2=a^2$$

(1)

$$\left(x^2+y^2\right)-2\left(px+qy\right)=a^2-p^2-q^2$$

(2)

$$r^2-2r\left(p\cos \theta +q\sin \theta \right)=a^2-p^2-q^2$$

(3)

$${\left(r-\left(p\cos \theta +q\sin \theta \right)\right)}^2=a^2-{\left(p\sin \theta -q\cos \theta \right)}^2$$

(4)

$$r=\left(p\cos \theta +q\sin \theta \right)\pm \sqrt{a^2-{\left(p\sin \theta -q\cos \theta \right)}^2}$$

(5)

If the origin position (0, 0) is inside the scanned log’s cross-section picture periphery and the radius (r) value is positive, then the positive root term at Equation (5) is a more suitable solution for the nonlinear regression model than the negative ones. Equation (6) is a nonlinear regression model to fit the circle’s radius with a center (p, q) and radius a, where r_i is the response variable, θ_i is the predictor variable, and ε_i is the residual. Equation (7) shows the circle’s center (p, q) lies at the (r₀, θ₀) polar coordinate, where p = r₀ cosθ₀ and q = r₀ sinθ₀. Equation (8) simplifies Equation (7):

$$r_i=\left(p\cos {\theta }_i+q\sin {\theta }_i\right)+\sqrt{a^2-{\left(p\sin {\theta }_i-q\cos {\theta }_i\right)}^2}+{\epsilon }_i$$

(6)

$$r_i=r_o\left(\cos {\theta }_o\cos {\theta }_i+\sin {\theta }_o\sin {\theta }_i\right)+\sqrt{a^2-r_o^2{\left(\sin {\theta }_i\cos {\theta }_o-\cos {\theta }_i\sin {\theta }_o\right)}^2}+{\epsilon }_i$$

(7)

$$r_i=r_o\cos \left({\theta }_i-{\theta }_o\right)+\sqrt{a^2-r_o^2{\left(\sin \left({\theta }_i-{\theta }_o\right)\right)}^2}+{\epsilon }_i$$

(8)

Iteration methods developed by the Gauss–Newton [45] or Levenberg–Marquardt [46,47] algorithms minimize the residual mean square to estimate the coefficient of regression parameters (a, r_o, and θ_o). The residual mean square, synonymous with the mean squared error (MSE), is the loss function that works acceptably for most regression cases. If r_i is the observed data point and ${\widehat{r}}_i$ is its estimated value, the residual square of the nonlinear regression model (Equation (6)) is ${\left(r_i-{\widehat{r}}_i\right)}^2$, and its MSE is Equation (9). Iterations of all parameter values are applied to find the least MSE. The initial parameter values, the starting point in the iteration process, should be carefully chosen because the MSE curve may have several peaks and valleys. The visual assessment of the data plot may significantly contribute to the well-chosen initial parameter values, which have a higher probability of resulting in the least MSE in the global range than those at the local valley.

$$MSE=\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{\left(r_i-{\widehat{r}}_i\right)}^2$$

(9)

2.2. Ellipse

The ellipse shape model is more flexible than the circular model; thus, its goodness of fit is usually improved. If the semi-major axis (a) and semi-minor axis (b) of an ellipse are the same values, the ellipse forms a circular shape. The standard formula of an ellipse with semi-major a, semi-major axis b, and center at coordinate (p, q) is presented in Equation (10). We arrange Equation (10) to become Equation (11), then substitute r cos θ to x, r sin θ to y, and r² to x² + y² to create Equation (12). Equation (12) is a quadratic equation; its solution is Equation (13). Equation (14) is another form of Equation (13). The k parameter is added, followed by the transformation of $p_1=q\sin k\pi +p\cos k\pi$ and $q_1=q\cos k\pi -p\sin k\pi$ (Equation (15)) to consider the rotation. The nonlinear curve-fitting model (Equation (16)) following the simplification of Equation (15) is chosen to fit the log’s cross-sectional shape and sapwood–heartwood transition that includes the rotation. Equation (17) is the nonlinear regression of an ellipse model without rotation because the kπ value is 0 (zero). Like the previous circular model, the nonlinear estimation can be employed following the iteration process with the least square loss function to calculate the best fit estimated parameters (a, b, r_o, θ_o, and k).

$$\frac{{\left(x-p\right)}^2}{a^2}+\frac{{\left(y-q\right)}^2}{b^2}=1$$

(10)

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{2px}{a^2}-\frac{2qy}{b^2}+\frac{p^2}{a^2}+\frac{q^2}{b^2}=1$$

(11)

$$\left(\frac{{\cos }^2\theta }{a^2}+\frac{{\sin }^2\theta }{b^2}\right)r^2-2\left(\frac{p\cos \theta }{a^2}+\frac{q\sin \theta }{b^2}\right)r+\left(\frac{p^2}{a^2}+\frac{q^2}{b^2}-1\right)=0$$

(12)

$$r_1,r_2=\frac{\left(\frac{p\cos \theta }{a^2}+\frac{q\sin \theta }{b^2}\right)\pm \sqrt{{\left(\frac{p\cos \theta }{a^2}+\frac{q\sin \theta }{b^2}\right)}^2-\left(\frac{{\cos }^2\theta }{a^2}+\frac{{\sin }^2\theta }{b^2}\right)\left(\frac{p^2}{a^2}+\frac{q^2}{b^2}-1\right)}}{\left(\frac{{\cos }^2\theta }{a^2}+\frac{{\sin }^2\theta }{b^2}\right)}$$

(13)

$$r_1,r_2=\frac{a^{2}q\sin \theta +b^{2}p\cos \theta \pm \sqrt{{\left(a^{2}q\sin \theta +b^{2}p\cos \theta \right)}^2-\left(a^2{\sin }^2\theta +b^2{\cos }^2\theta \right)\left(a^2{q}^2+b^2{p}^2-a^2{b}^2\right)}}{a^2{\sin }^2\theta +b^2{\cos }^2\theta }$$

(14)

$$r=\frac{a^2\left(q\cos k\pi -p\sin k\pi \right)\sin \left(\theta -k\pi \right)+b^2\left(q\sin k\pi +p\cos k\pi \right)\cos \left(\theta -k\pi \right)+\sqrt{{\left(a^2\left(q\cos k\pi -p\sin k\pi \right)\sin \left(\theta -k\pi \right)+b^2\left(q\sin k\pi +p\cos k\pi \right)\cos \left(\theta -k\pi \right)\right)}^2-\left(\left(a^2-b^2\right){\left(\sin \left(\theta -k\pi \right)\right)}^2+b^2\right)\left(a^2{\left(q\cos k\pi -p\sin k\pi \right)}^2+b^2{\left(q\sin k\pi +p\cos k\pi \right)}^2-a^2{b}^2\right)} }{\left(a^2-b^2\right){\left(\sin \left(\theta -k\pi \right)\right)}^2+b^2}$$

(15)

$${\widehat{r}}_i=\frac{r_o\left(\left(a^2-b^2\right)\sin \left({\theta }_i-k\pi \right)\sin \left({\theta }_o-k\pi \right)+b^2\cos \left({\theta }_i-{\theta }_o\right)\right)\pm ab\sqrt{\left(a^2-b^2\right){\sin }^2\left({\theta }_i-k\pi \right)+b^2-r_o{}^2{\sin }^2\left({\theta }_i-{\theta }_o\right)}}{\left(a^2-b^2\right){\sin }^2\left({\theta }_i-k\pi \right)+b^2}+{\epsilon }_i$$

(16)

$${\widehat{r}}_i=\frac{r_o\left(\left(a^2-b^2\right)\sin {\theta }_i\sin {\theta }_o+b^2\cos \left({\theta }_i-{\theta }_o\right)\right)\pm ab\sqrt{\left(a^2-b^2\right){\sin }^2{\theta }_i+b^2-r_o{}^2{\sin }^2\left({\theta }_i-{\theta }_o\right)}}{\left(a^2-b^2\right){\sin }^2{\theta }_i+b^2}+{\epsilon }_i$$

(17)

2.3. Superellipse

A superellipse, introduced by Gabriel Lamé [48], is a closed curved where its plot in a rectangular diagram generally follows Equation (18). The superellipse equation is adaptive, embracing circular, ellipse, square, and rectangular shapes. The step-by-step algebraic procedure in Equations (19)–(20) changes the superellipse curve plot centered at (0, 0) at the rectangular diagram (Equation (18)) to be plotted at the polar diagram (Equation (21)), and then Equation (22) determines the rotated superellipse at a polar diagram. Nonlinear regression can be applied based on the superellipse model (Equations (23) and (24)) following similar procedures with the above circular and ellipse model parameters’ estimation.

$${\left|\frac{x}{a}\right|}^n+{\left|\frac{y}{b}\right|}^n=1$$

(18)

$${\left|\frac{\left(r\cos \theta \right)}{a}\right|}^n+{\left|\frac{\left(r\sin \theta \right)}{b}\right|}^n=1$$

(19)

$$r^n\left({\left|\frac{\left(\cos \theta \right)}{a}\right|}^n+{\left|\frac{\left(\sin \theta \right)}{b}\right|}^n\right)=1$$

(20)

$$r=\frac{ab}{\sqrt[n]{{\left|b\left(\cos \theta \right)\right|}^n+{\left|a\left(\sin \theta \right)\right|}^n}}$$

(21)

$$r=\frac{ab}{\sqrt[n]{{\left|b\left(\cos \theta -k\pi \right)\right|}^n+{\left|a\left(\sin \theta -k\pi \right)\right|}^n}}$$

(22)

$$r_i=\frac{ab}{\sqrt[n]{{\left|b\left({\cos \theta }_i\right)\right|}^n+{\left|a\left({\sin \theta }_i\right)\right|}^n}}+{\epsilon }_i$$

(23)

$$r_i=\frac{ab}{\sqrt[n]{{\left|b\left({\cos \theta }_i-k\pi \right)\right|}^n+{\left|a\left({\sin \theta }_i-k\pi \right)\right|}^n}}+{\epsilon }_i$$

(24)

2.4. Computer Modelling

2.4.1. Python

Python programming language is a high-level programming software [49,50] to develop data science and machine learning created by Guido Van Rossum, a programmer from the Netherlands [49,50,51,52]. The Python programming language has several elements, such as syntax [53], the variable of which is a value or data [54,55] used in Python programming, data structure [54,56], functions in performing data processing [51], modules consisting of combining functions and programming code, libraries containing data science and machine learning [57], interpreted [49,53,54], and cross-platform that can support Python on several platforms [51]. In this study, Python was employed to develop a rotated ellipse graph. Building a program using Python programming language [58] generally includes structures such as headers, comments, import statements, function definitions, and main programs [59] as a basis which can later be developed into a special format intended for the program to be created. The general structures of Python code are:

• Header or shebang line (optional): on Unix/Linux/Mac platforms, this line tells the shell that the file should be interpreted using the Python interpreter.
• Comments (optional): this section provides comments or explanations about the code being written, but Python does not execute it.
• Import statement (optional): this section imports modules used in the program.
• Function definition (optional): in this section, we can define functions that will be used in the program.
• Main program: in this section, we write the main code of the program that will be executed.

2.4.2. Qt

Qt is a cross-platform application development framework that provides a variety of components, tools, and programming languages to build applications [60]. Qt aims to simplify and accelerate the application development process by offering a consistent and robust framework [61]. It also focuses on the ease of designing attractive and responsive user interfaces to enhance user experience. In summary, Qt is a versatile application development framework that simplifies the cross-platform development process and designs attractive user interfaces. It offers benefits such as cross-platform capability, attractive user interfaces, high productivity, modular architecture, active community support, and commercial support options [62].

3. Materials and Methods

A chainsaw-man team cut a Maesopsis eminii tree from the Arboretum of Forestry and Environment Faculty, IPB University, Bogor, West Java (ID). Then, they cut the tree into several logs. The photographs of the randomly chosen logs’ cross-sections were taken and then scaled into the actual dimension (1:1). The points of the log’s circumference and the sapwood–heartwood transition periphery were digitized and plotted on a rectangular diagram. The data points count ranged from 80 to 150 for each periphery curve. Equations (25)–(27) transformed every data point’s rectangular coordinate (x, y) into a polar coordinate (r, θ). Since Microsoft Excel’s “=asin (number)” formula results in the value range of −π/2 ≤ θ ≤ π/2, Equation (28) adjusted the θ value into its point position at the 1st, 2nd, 3rd, or 4th quadrant; for example, when the cell B2 contains x, and cell C2 contains y; Equation (28) is a formula to define the quadrant where the point lays. Equation (29) is the formula to determine the angle (θ), where cell F2 contains the quadrant resulting from Equation (28), and cell E2 contains “=sin θ” resulting from Equation (26). The radius (r) unit is cm, while the angle (θ) unit is radian.

$$r=\sqrt{x^2+y^2}$$

(25)

$$\sin \theta =\frac{y}{r}$$

(26)

$$\theta =\{\begin{array}{c}\begin{array}{l}\mathrm{asin}\left(\frac{y}{r}\right);\mathrm{for} \left\{0 \le x \mathrm{and} 0 \le y\right\}\hfill \\ \pi -\mathrm{asin}\left(\frac{y}{r}\right);\mathrm{for} \left\{0>x \mathrm{and} 0<y\right\}\hfill \end{array}\\ \begin{array}{l}\pi -\mathrm{asin}\left(\frac{y}{r}\right);\mathrm{for} \left\{0>x \mathrm{and} 0>y\right\}\hfill \\ 2\pi +\mathrm{asin}\left(\frac{y}{r}\right); \mathrm{for} \left\{0<x \mathrm{and} y<0\right\} \hfill \end{array}\end{array}$$

(27)

=IF(AND(B2>=0,C2>=0),1,IF(AND(B2<0,C2>0),2,IF(AND(B2<0,C2<0),3,4)))

(28)

=IF(F2=1,ASIN(E2),IF(F2=2,PI()-ASIN(E2),IF(F2=3,PI()-ASIN(E2),2*PI()+ASIN(E2))))

(29)

All raw data points’ polar coordinates (r_i, θ_i) were tabulated on a spreadsheet. Statsoft Statistica 12, a nonlinear estimation of advanced linear/nonlinear models, was employed to estimate the (a, b, r_o, θ_o, and k) parameters in Equations (8), (16), (17) and (24); the loss function is the least square, and the iteration method is Levenberg–Marquardt. We chose the 10⁻¹⁰ maximum convergence criteria, 10,000 maximum iterations, and varied starting values. Visual assessment observed the raw data plot, and the Desmos calculator initially fitted the curve of the data plot. The Desmos graph, which nearly fits the raw data points, contributes to choosing the initial parameter values for the iteration starting values.

The model’s goodness of fit was evaluated using the coefficient of determination (R²), which indicated the proportion of the accounted variance. The higher value of R² leads to a better-fit model. We also calculated the root of mean square error (RMSE) as the model’s goodness of fit measurement, whose smaller value indicates the better model. The observed and estimated data are plotted with the x = y line. The t-values measure the significant relationship between radius (r) and angle (θ), where the hypothesis represents the non-zero value of estimated parameters (a, b, r_o, θ_o, and k). The null hypothesis concerning this model are H0: a = 0, b = 0, r_o = 0, θ_o = 0, and k = 0, while the alternative hypothesis H1: a ≠ 0, b ≠ 0, r_o ≠ 0, θ_o ≠ 0, and k ≠ 0. Since we determined the 95% confidence level, the null hypothesis is rejected, and the alternative hypothesis is accepted if the |t-value| > t_0.025 or the p-value < 0.025.

The basic pattern model algorithm transforms the provision of formulas from ellipse formulas into computer models using Python scripts provided by libraries from the Wolfram engine. Variable declaration, initialization, process, and output by applying the formula are below:

4. Results and Discussions

4.1. Circle Model

Figure 1 shows a nonlinear regression curve fitting for all pairs of angles (θ) and radius (r) data points following the circular model (Equation (8)) with medium to high precision (R² = 0.4686 for the log cross-section outer circumference and R² = 0.8910 for sapwood–heartwood transition). The least-square loss function, which runs following the Levenberg–Marquardt iteration algorithm [63,64], could produce reliable parameter estimates.

Table 1. The best fit estimated parameters of the outer log circumference (a) and the sapwood–heartwood transition (b) curve based on the circle model in Equation (8).

Parameter	Estimate	Standard Error	t-Value	p-Value	95% Lower Conf. Limit	95% Upper Conf. Limit
a. Outer log circumference
ro	0.2099	0.0215	9.7612	0.0000	0.1672	0.2525
θo	2.2891	0.1040	22.0127	0.0000	2.0830	2.4952
a	8.8597	0.0154	577.0298	0.0000	8.8292	8.8901
b. Sapwood–Heartwood transition
ro	−0.7447	0.0305	−24.4118	0.0000	−0.8054	−0.6839
θo	−2.1697	0.0396	−54.8502	0.0000	−2.2486	−2.0909
a	5.2821	0.0212	248.6318	0.0000	5.2397	5.3244

summarizes the circle curve’s estimated parameters (a, r_o, and θ_o) based on the Equation (6) model, which best fits the sapwood and heartwood transition.

Table 2. The analysis of variance of the outer log circumference (a) and the sapwood–heartwood transition (b) curve based on the circle model in Equation (8).

Effect	Sum Square	df	Mean Square	F-Value	p-Value
a. Outer log circumference
Regression	8666.97	3	2888.99	112,537.32	0.0000
Residual	2.75	107	0.03
Total	8669.71	110
Corrected Total	5.17	109
Regression vs. Corrected Total	8666.97	3	2888.99	60,924.84	0.0000
b. Sapwood–Heartwood transition
Regression	2151.38	3	717.13	20,682.72	0.0000
Residual	2.57	74	0.03
Total	2153.95	77
Corrected Total	23.54	76
Regression vs. Corrected Total	2151.38	3	717.13	2315.32	0.0000

summarizes the analysis of variance (ANOVA). All estimated parameters have a p-value less than 0.025, and their |t-value| is more than t_0.025; thus, the null hypothesis is rejected, and the alternative hypothesis is accepted. The sapwood and heartwood transition geometric shapes are circular and have radius r_i and center coordinates (r_o, θ_o) (Figure 1c).

Table 1, b section, shows the significant estimated parameter values which form the curve of circular shapes having 5.28 cm and 8.86 radii. The heartwood center’s polar coordinate (−0.74 cm, −2.17 rad) and the log center’s polar coordinate (0.21 cm, 2.29 rad) are not at the same points, and they are not at the pith position (Figure 1c). The different center positions indicate that the sapwood–heartwood formation does not follow the tree’s annual ring and strengthen Bamber’s report [65] that the heartwood formation is the product of the developmental process rather than a deterioration in cell function with age. The sapwood–heartwood color transition is sometimes unclear, and it must be identified based on the fundamental change, such as the death of the vertical parenchyma and the ray cells [66]. The center shift phenomenon is also consistent with Taylor et al.’s hypothesis that the amount of sapwood converted into heartwood is required to balance the foliar and energy demands [67].

A circular curve-fitting following the Equation (6) model successfully idealizes the sapwood–heartwood transition (Figure 1). The high coefficient of the determination value (R² = 0.8910) confirms that the high proportion of variable variation in the sapwood–heartwood transition radius (r_i) can be predictable from the angle (θ). This R² value of sapwood–heartwood transition regression is even higher than the R² of the log cross-section outer periphery regression. The low MSE value also emphasizes the circular curve’s high goodness of fit (Table 2, b section). This circular model, therefore, is sufficiently reliable to be advocated for determining the sapwood–heartwood transition. This circular model can estimate the sapwood and heartwood area, and then the heartwood to sapwood proportion can also be calculated.

This study justifies that the shape simplification following the nonlinear circle model can fit the outer log periphery well enough. This developing method provides a precise quantitative number to determine a log’s idealized cross-sectional shape and size. Table 1, a section, shows the estimated parameter values, which form the best-fit circular curve to the raw data. The more convincing goodness of fit, demonstrated by the high coefficient of determination (R²) and small root mean square error (RMSE) value, improves the forester’s confidence in continuing using the circular shape assumption to measure the log’s cross-sectional area. The coefficient of determination (R²) indicates the proportion of variance accounted for is 0.469 for the log cross-section outer periphery. Table 2 shows the mean square error is 0.03 (its root is 0.16), much smaller than the mean square corrected total.

The deviation from a circular form in the stem’s cross-section shape may reasonably be ignored. The bias between circular and ellipse shapes has long been considered small [18]. Researchers have chosen a circle as the geometric shape of the log’s cross-section and tree-ring periphery for more than 75 years in multidisciplinary ecological research. The circular basal area of the tree is traditionally calculated in practice based on the diameter at breast height (D_bh) [68,69,70,71]. D_bh is generally the average value of the maximum diameter (major axis) and the minimum diameter (minor axis) at 1.3 m height.

As stated before, the circular shape is sufficient to fit the outer periphery of a log. However, the tree often physiologically adjusts the cambium cell differentiation rates during the xylem forming as a reaction of the physiological responses (i.e., intercellular signaling [72], sucrose signaling [73], long non-coding RNAs (lncRNAs) [74], the receptor-like kinase PXY and its ligand CLE41 [75], transcriptional regulator AHL15 [76],) to the local environmental stress [77,78]; thus, the variation in the outer log circumference may exist. Some alternative geometric shape models, such as ellipse and superellipse, are also discussed in this study to fit the variation in a log’s cross-section shape and sapwood–heartwood transition.

4.2. Ellipse Model

A tree stem’s diameter at the top (D_t) is usually smaller than the diameter at breast height (D_bh). Its three-dimensional shape may resemble a truncated cone [79], sometimes called a conical frustum, as the best-fit idealized standard geometric form. The tree’s or log’s tapered stem volume equations, empirically proposed by foresters, are Equations (30)–(34) [80]. Equation (31) is the best-known formula, where b₁ denotes the breast-height form factor. Those empirical equations may slightly deviate from the exact formula for calculating the volume of an idealized conical frustum (Equation (35a)). If the ratio of the top diameter per the bottom diameter (D_t/D_b) is defined as a taper (1/α), its substitution to Equation (35a) results in Equation (35b). An individual tree stem’s taper is assumed to be constant; therefore, the breast-height form factor (b₁) is a constant that expresses π(1 + α² + α)/12 and D_b transformation to D_bh. Equation (35c) agrees with Equation (30) and generally confirms that foresters had recognized a conical frustum shape that matched the trunk shape of an individual tree.

$$V=b_0+b_1{D}^{2}h$$

(30)

$$V=b_1{D}^{2}h$$

(31)

$$V=b_1{D}^{b_2}{h}^{b_3}$$

(32)

$$V=b_0+b_1{D}^{b_2}{h}^{b_3}$$

(33)

$$V=\frac{D^2}{b_0+\frac{b_1}{h}}$$

(34)

$$V=\frac{\pi }{12}\left(D_b^2+D_t^2+D_b{D}_t\right)h$$

(35a)

$$V=\frac{\pi }{12}\left(D_b^2+{\alpha }^2{D}_b^2+\alpha D_b^2\right)h$$

(35b)

$$V=\frac{\pi }{12}\left(1+{\alpha }^2+\alpha \right)D_b^{2}h$$

(35c)

where V = stem volume; D = diameter at the breast height (D_bh); D_t = diameter at the top; D_b = diameter at the bottom; h = tree height or log length; b₀, b₁, b₂, and b₃ = estimated regression coefficients; 1/α = taper.

Slicing a conic section may result in a circle, ellipse, parabolic, or hyperbolic plane. Cutting a tree’s stem, which resembles a conical frustum, into several logs may result in a circle or ellipse cross-sectional shape. A perpendicular to the longitudinal axis cut resulted in a circle, while an oblique cut resulted in an ellipse cross-section. There is a higher probability that the chainsaw man or harvester made an oblique cut, resulting in an ellipse cross-sectional shape. Therefore, elliptical cross-section logs are more numerous, prevalent, and easier to find than circular ones.

In addition to the indication of an oblique cut, the better-fit ellipse curve may show the environmental factors affecting the secondary growth of the tree. The resultant of wind, snow, and unsymmetric crown self-weight generate the non-eccentric compressive load the stem receives. Their combination with the bending moment triggers buckling, severely reducing its load-bearing capacity [79,81,82,83,84,85]. The buckling generates compressive stress in half of the stem and tensile stress in the other half. A tree’s stem reacts to the gradual stress by adjusting the cambium cell differentiation rates, producing the reaction wood, and may form an almost ellipse shape in cross-section. The gradual stress and strain in the tree stem give rise to long-duration tensile or compressive loading, and creep and relaxation directly affect the vascular cambium [86].

An inclined tree produces reaction wood and distorts the circularly uniform cambium differentiation. The cambiums of a buckled or inclined tree have a sloping orientation to the normal vertical direction; thus, the gravitational stimulus influences the auxin hormone distribution and creates the abnormal woody tissue, namely reaction wood. Gymnosperms produce the reaction wood on the lower side of leaning stems, called compression wood; meanwhile, a dicotyledonous angiosperms’ reaction wood is formed on the leaning stem’s upper side and is referred to as tension wood [87,88,89]. The non-uniform growth rate of reaction wood places the pith rather to the edge, and the cross-section deviates from the circular to become more similar to an ellipse than a circle. The competition for light, space, water, and nutrition significantly affects the xylem cell differentiation [90,91] and also contributes to deviating from a perfectly circular cross-section.

Because of its more adaptive parameter, the ellipse model resulted in a better-fit shape to a log cross-section than a circle. A circle is a special form of an ellipse if the major axis has the same value as the minor axis. The curve would form a circle if the nonlinear regression following an ellipse model resulted in similar values of semi-major (a) and semi-minor (b). Figure 2(a1,b1,c1) is the graph of the ellipse curve-fit result following the Equation (17) model, while Figure 2(a2,b2,c2) is the graph of the rotated ellipse curve-fit result following the Equation (16) model.

Table 3. The best fit estimated parameters of the outer log circumference (a) and the sapwood–heartwood transition (b) curve-fit based on the ellipse model in Equation (17).

Parameter	Estimate	Standard Error	t-Value	p-Value	95% Lower Conf. Limit	95% Upper Conf. Limit
a. Outer log circumference
ro	0.208	0.021	10.1510	0.0000	0.1676	0.2489
a	8.788	0.025	350.7989	0.0000	8.7388	8.8381
b	8.933	0.026	345.4442	0.0000	8.8819	8.9844
θo	2.293	0.100	22.9810	0.0000	2.0948	2.4903
b. Sapwood–Heartwood periphery
ro	0.748	0.029	25.5898	0.0000	0.690	0.807
a	5.197	0.035	147.6168	0.0000	5.127	5.268
b	5.366	0.035	151.1844	0.0000	5.295	5.436
θo	0.979	0.037	26.2172	0.0000	0.904	1.053

summarizes the estimated parameters of the ellipse model, while

Table 4. The best fit estimated parameters of the outer log circumference (a) and the sapwood–heartwood transition (b) curve-fit based on the rotated ellipse model in Equation (16).

Parameter	Estimate	Standard Error	t-Value	p-Value	95% Lower Conf. Limit	95% Upper Conf. Limit
a. Outer log circumference
ro	−0.1918	0.0158	−12.1445	0.0000	−0.2231	−0.1605
a	8.7051	0.0187	465.8943	0.0000	8.6680	8.7421
b	9.0159	0.0192	470.5462	0.0000	8.9779	9.0539
k	2.1767	0.0161	135.1281	0.0000	2.1448	2.2086
θo	−0.8796	0.0803	−10.9550	0.0000	−1.0388	−0.7204
b. Sapwood–Heartwood transition
ro	−0.7403	0.0014	−526.8547	0.0000	−0.7431	−0.7375
a	5.0235	0.0017	2932.1950	0.0000	5.0200	5.0269
b	5.5353	0.0018	3122.5152	0.0000	5.5318	5.5388
k	2.1943	0.0009	2462.1008	0.0000	2.1926	2.1961
θo	−2.1344	0.0020	−1083.8741	0.0000	−2.1384	−2.1305

summarizes those of the rotated model. As expected, the ellipse model significantly improves the goodness of fit, shown by the increasing coefficient of determination (R²) value. The ellipse model’s R² values are 0.523 and 0.902 for fitting the log cross-section shape and sapwood–heartwood transition, respectively, while those of the rotated ellipse are 0.7317 and 0.9998. A rotated ellipse better fits the raw data than the non-rotated one. Both ellipse models’ goodness of fit is higher than the circular model. The analysis of variances of the ellipse (

Table 5. The analysis of variance of the outer log circumference (a) and the sapwood–heartwood transition (b) curve-fit based on the ellipse model in Equation (17).

Effect	Sum Square	df	Mean Square	F-Value	p-Value
a. Outer log circumference
Regression	8667.25	4	2166.81	93077.35	0.0000
Residual	2.47	106	0.02
Total	8669.71	110
Corrected Total	5.17	109
Regression vs. Corrected Total	8667.25	4	2166.81	45695.10	0.0000
b. Sapwood–heartwood transition
Regression	2151.65	4	537.91	17081.85	0.0000
Residual	2.30	73	0.03
Total	2153.95	77
Corrected Total	23.54	76
Regression vs. Corrected Total	2151.65	4	537.91	1736.71	0.0000

) and the rotated ellipse (

Table 6. The analysis of variance of the outer log circumference (a) and the sapwood–heartwood transition (b) curve based on the rotated ellipse model in Equation (16).

Effect	Sum Square	df	Mean Square	F-Value	p-Value
a. Outer log circumference
Regression	8668.33	5	1733.67	131,252.79	0.0000
Residual	1.39	105	0.01
Total	8669.71	110
Corrected Total	5.17	109
Regression vs. Corrected Total	8668.33	5	1733.67	36,560.64	0.0000
b. Sapwood–heartwood transition
Regression	2153.94	5	430.79	5,463,867.57	0.0000
Residual	0.01	72	0.00
Total	2153.95	77
Corrected Total	23.54	76
Regression vs. Corrected Total	2153.94	5	430.79	1390.85	0.0000

) curve fitting models also indicate the goodness of fit enhancement. The MSE values of both ellipse models are smaller than that of the circle model.

Similar to the circle model’s results, the log and sapwood–heartwood transition centers are in different locations. The rotated ellipse model estimated that the log’s center is (0.208 cm, 2.293 rad) and the heartwood center is (0.748 cm, 0.979 rad). Meanwhile, the non-rotated ellipse model resulted in (−0.1918 cm, −0.8796 rad) and (−0.7403 cm, −2.1344 rad) polar coordinate locations for log and heartwood center, respectively. The sapwood–heartwood transition did not coincide with a tree ring.

4.3. Superellipse Model

If the superellipse curve’s center is the origin (0, 0), its formula at the rectangular diagram is Equation (18). Converting the rectangular diagram to a polar diagram, r cos θ substitution to x, and r sin θ to y, result in Equation (19). Equation (22) includes the kπ angle rotation into the superellipse formula. Applying nonlinear regression to fit the superellipse models (Equations (23) and (24)) resulted in a less precise curve than the ellipse or circular models. Figure 3 indicates the systematic error of the prediction curve compared to the original data. This unsatisfactory result may appear because the superellipse’s center is assumed at the origin (0, 0). The solution model needs to find the best-fit estimated superellipse’s central location. Since the nonlinear regression of the superellipse is unreliable enough to determine the best-fit parameters (

Table 7. The best fit estimated parameters of the sapwood (a) and heartwood (b) periphery curve-fit based on the superellipse model in Equation (23).

Parameter	Estimate	Standard Error	t-Value	p-Value	95% Lower Conf. Limit	95% Upper Conf. Limit
a. Outer log circumference
a	8.8362	0.0483	182.7632	0.0000	8.7404	8.9320
n	1.9574	0.0370	52.9616	0.0000	1.8841	2.0306
b	8.9933	0.0504	178.3683	0.0000	8.8933	9.0932
b. Sapwood–Heartwood transition
a	5.1503	0.1509	34.1304	0.0000	4.8497	5.4510
n	2.0653	0.2201	9.3855	0.0000	1.6269	2.5038
b	5.3045	0.1537	34.5169	0.0000	4.9983	5.6107

and

Table 8. The best fit estimated parameters of the sapwood (a) and heartwood (b) periphery curve-fit based on the rotated superellipse model in Equation (24).

Parameter	Estimate	Standard Error	t-Value	p-Value	95% Lower Conf. Limit	95% Upper Conf. Limit
a. Outer log circumference
k	1.11	0.02	69.64	0.0000	1.08	1.14
a	8.74	0.04	236.25	0.0000	8.66	8.81
n	2.29	0.05	48.12	0.0000	2.19	2.38
b	8.53	0.07	123.70	0.0000	8.39	8.66
b. Sapwood–Heartwood transition
k	−0.81	0.07	−12.3972	0.0000	−0.94	−0.68
a	5.05	0.15	34.1796	0.0000	4.76	5.35
n	1.99	0.20	10.1658	0.0000	1.60	2.37
b	5.48	0.15	37.0000	0.0000	5.18	5.77

), justified by the lower analysis of variance (

Table 9. The analysis of variance of the outer log circumference (a) and the sapwood–heartwood transition (b) curve-fit based on the superellipse model in Equation (23).

Effect	Sum Square	df	Mean Square	F-Value	p-Value
a. Outer log circumference
Regression	8664.92	3	2888.31	64511.86	0.0000
Residual	4.79	107	0.05
Total	8669.71	110
Corrected Total	5.17	109
Regression vs. Corrected Total	8664.92	3	2888.31	60910.48	0.0000
b. Sapwood–heartwood transition
Regression	2130.67	3	710.22	2257.49	0.0000
Residual	23.28	74	0.31
Total	2153.95	77
Corrected Total	23.54	76
Regression vs. Corrected Total	2130.67	3	710.22	2293.03	0.0000

and

Table 10. The analysis of variance of the outer log circumference (a) and the sapwood–heartwood transition (b) curve-fit based on the rotated superellipse model in Equation (24).

Effect	Sum Square	df	Mean Square	F-Value	p-Value
a. Outer log circumference
Regression	8666.23	4	2166.56	65930.26	0.0000
Residual	3.48	106	0.03
Total	8669.71	110
Corrected Total	5.17	109
Regression vs. Corrected Total	8666.23	4	2166.56	45689.75	0.0000
b. Sapwood–heartwood transition
Regression	2132.14	4	533.04	1784.61	0.0000
Residual	21.80	73	0.30
Total	2153.95	77
Corrected Total	23.54	76
Regression vs. Corrected Total	2132.14	4	533.04	1720.96	0.0000

), we propose another method to accommodate the curve-fitting of the superellipse curve. Intrinsic regression can be employed to fit the superellipse models, and we will discuss it in a future study.

4.4. Computer Modelling

The rotated ellipse model is the best-fit model, and it also covers a circle and a regular ellipse. Therefore, this study chooses to develop computer programming for the rotated ellipse model. The executable program script is below, and Figure 4 displays its user interface.

Figure 4. User interface display of computer programming for a rotated ellipse graph at the polar diagram system.

Figure 4. User interface display of computer programming for a rotated ellipse graph at the polar diagram system.

5. Conclusions

This study discusses transforming the rectangular coordinate system into a polar coordinate system to solve the non-function of the log’s cross-section shape and sapwood–heartwood periphery. This transformation successfully solves the difficulty of non-function and paves the way to function; thus, the nonlinear regression methods smoothly solve it. The transformation of raw data points’ positions of a closed curve from the rectangular coordinate (x, y) into the polar coordinates (r, θ) raises the possibility of fitting the log circumference and sapwood–heartwood transition using nonlinear regression. The circular shape model is sufficient to fit the outer periphery of the logs and sapwood–heartwood transition, while ellipse models significantly improve their goodness of fit. The rotated ellipse model is the best fit among others. The deviation from the circular shape indicates the environmental effect on vascular cambium differentiation. Foresters have good choices: (1) continue using the circular model as the simplest one, or (2) change to the rotated ellipse model because it is the best fit to estimate the tree stem’s cross-sectional shape; therefore, it can more precisely determine the basal area, tree volume, and tree trunk biomass. This study developed computer modelling for the rotated ellipse graph in the polar diagram system using Python scripts provided by libraries from the Wolfram engine.