The conic-paraboloid volume equation is receiving increased use with downed coarse woody material (CWM), but the consequences for taper have not been identified mathematically. Requiring that subdivision of a conic-paraboloid yields two smaller conic-paraboloids leads to an exact taper equation intermediate between those of cones and second-order paraboloids. This exact taper equation does not have an explicit inverse, however. An alternative, naive approach does have an explicit inverse, but subdivision does not yield two conic-paraboloids. The exact conic-paraboloid is closely approximated by Fermat’s paraboloid with exponent 7/5. The exact and naive conic-paraboloids match in volume; differences in taper are ≤2.2% of large-end cross-sectional area and ≤5.9% of large-end diameter, while differences in inverse taper are ≤3.7% of total length. Fermat’s paraboloid is always within 1.2% of total volume; differences in taper are ≤0.8% of large-end cross-sectional area and ≤2.0% of large-end diameter, while differences in inverse taper are ≤1.1% of total length. Such differences are negligible given the variety of CWM shapes and practical measurement challenges. Either the exact conic-paraboloid or the corresponding Fermat’s paraboloid provides appropriate equations for estimating the volume and taper of CWM that is intermediate between conical and ordinary paraboloid frusta.
ABSTRACT
PUBLICATION RECORD
- Publication year
2018
- Venue
Canadian Journal of Forest Research
- Publication date
2018-05-23
- Fields of study
Mathematics, Environmental Science
- Identifiers
- External record
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Semantic Scholar
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