Superellipse Equation Describing the Geometries of <i>Abies alba</i> Tree Rings
Our previous study using 41 tree rings of one <i>Abies alba</i> Mill. disc indicated that the superellipse equation can accurately fit its tree-ring shape. This study further used the superellipse equation (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML&quo...
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| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
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| Series: | Plants |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2223-7747/13/24/3487 |
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| Summary: | Our previous study using 41 tree rings of one <i>Abies alba</i> Mill. disc indicated that the superellipse equation can accurately fit its tree-ring shape. This study further used the superellipse equation (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>x</mi></mrow><mrow><mi>a</mi></mrow></mfrac></mstyle></mrow></mfenced></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>y</mi></mrow><mrow><mi>β</mi></mrow></mfrac></mstyle></mrow></mfenced></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> ) to model the geometries of 1090 tree rings of <i>A. alba</i> discs collected from five sites in Denmark. The adjusted root-mean-square-error (RMSE<sub>adj</sub>) was calculated to assess the goodness of fit between observed and predicted tree-ring boundaries. The results showed that RMSE<sub>adj</sub> ranged between 0.0038 and 0.0591, with a mean value of 0.0141. This verified that the superellipse equation sufficiently describes the <i>A. alba</i> tree-ring shape. In the polar coordinate system, the superellipse equation can be expressed as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>=</mo><msup><mrow><mi>a</mi><mo>(</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>φ</mi></mrow></mfenced></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><msup><mrow><mfenced open="|" close="|" separators="|"><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>φ</mi><mo>/</mo><mi>k</mi></mrow></mfenced></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mi>n</mi></mrow></msup></mrow></semantics></math></inline-formula>. Where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi></mrow></semantics></math></inline-formula> are the polar radius and polar angle, respectively. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mi>β</mi><mo>/</mo><mi>a</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi></mrow></semantics></math></inline-formula> are the major and minor semi-axes of the superellipse. The mean value of <i>k</i> was 0.95, 94% of tree rings had <i>k</i>-values between 0.90 and 1.00, and only 67 tree rings had <i>k</i>-values between 0.71 and 0.90. <i>n</i>-value ranged from 1.62 to 2.81, with an average value of 2.04. 59% of the tree rings had <i>n</i>-values between 1.90 and 2.10, and 62% showed <i>n</i>-values greater than 2.0. This means that most tree rings are a hyperellipse approached to an ellipse. Sites with different soil moisture conditions influenced the size but not the shape of tree rings. This study verified that the tree-ring shape of <i>A. alba</i> tends to be bilaterally symmetric and hyperellipse approached ellipse. Its variation was reflected more in inter-annual differences in <i>k</i>- and <i>n</i>-values. |
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| ISSN: | 2223-7747 |