The Natural 3D Spiral

Harary, Gur; Tal, Ayellet

doi:10.1111/j.1467-8659.2011.01855.x

Cited by 49 publications

(37 citation statements)

References 29 publications

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“…That opens the door to further extensions: the same methodology could be applied to get implicit primitives along 3D spiral skeletons, which were shown to be very useful to represent a larger variety of natural shapes such as seashells and horns [HT11]. Modifying our deformation to the sweeping of more general cross-sections around the helix skeleton would also be a useful extension of this work, made possible by our section preserving warp.…”

Section: Discussionmentioning

confidence: 93%

Warp-based helical implicit primitives

Zanni

Hubert

Cani

2011

Computers & Graphics

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Implicit modeling with skeleton-based primitives has been limited up to now to planar skeletons elements, since no closed-form solution was found for convolution along more complex curves. We show that warping techniques can be adapted to efficiently generate convolution-like implicit primitives of varying radius along helices, a useful 3D skeleton found in a number of natural shapes. Depending on a single parameter of the helix, we warp it onto an arc of circle or onto a line segment. For those latter skeletons closed form convolutions are known for entire families of kernels. The new warps introduced preserve the circular shape of the normal cross section to the primitive.

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Section: Discussionmentioning

confidence: 93%

Warp-based helical implicit primitives

Zanni

Hubert

Cani

2011

Computers & Graphics

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“…Picado, 2009, Stępień, 2009Meinhardt, 2009;Urdy et al, 2010;Harary & Tal, 2011;Moulton, Goriely & Chirat, 2012;Faghih Shojaei et al, 2012;Chacon, 2012). Here, we will not further discuss the details of the at least 29 published shell models, but refer to the comprehensive overviews and descriptions of these models in Dera et al (2009) and Urdy et al (2010).…”

Section: Preprintsmentioning

confidence: 99%

“…We estimated two differential geometry parameters, namely, curvature (κ) torsion (τ), and ontogeny axis length for all apertures (Okamoto, 1988;Harary & Tal, 2011). The local curvature and torsion, and accumulative ontogeny axis length were estimated from the aperture points along the growth trajectory by using weighted least-squares fitting and local arc length approximation (Lewiner et al, 2005).…”

Section: Quantifying Aperture Growth Trajectorymentioning

confidence: 99%

A method for quantifying, visualising, and analysing gastropod shell form

Liew

2014

Preprint

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Quantitative analysis of organismal form is an important component for almost every branch of biology. Although generally considered an easily-measurable structure, the quantification of gastropod shell form is still a challenge because shells lack homologous structures and have a spiral form that is difficult to capture with linear measurements. In view of this, we adopt the idea of theoretical modelling of shell form, in which the shell form is the product of aperture ontogeny profiles in terms of aperture growth trajectory that is quantified as curvature and torsion, and of aperture form that is represented by size and shape. We develop a workflow for the analysis of shell forms based on the aperture ontogeny profile, starting from the procedure of data preparation (retopologising the shell model), via data acquisition (calculation of aperture growth trajectory, aperture form and ontogeny axis), and data presentation (qualitative comparison between shell forms) and ending with data analysis (quantitative comparison between shell forms). We evaluate our methods on representative shells of the genus Opisthostoma and Plectostoma, which exhibit great variability in shell form. The outcome suggests that our method is robust, reproducible, and versatile for the analysis of shell form. Finally, we propose several potential applications of our methods in functional morphology, theoretical modelling, taxonomy, and evolutionary biology.PeerJ PrePrints | http://dx.doi.org/10.7287/peerj.preprints.157v2 | CC-BY 3.0

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“…After that, we calculated the distance of the portion of the whole predatory path which corresponded to the predatory path for each the 11 growth stages, and plotted these predatory path distances on the ontogeny axis ( Figure 4E). Then, we described the geometry of the shell whorls as a 3D spiral, in the terms of torsion and radius of curvature (Harary & Tal, 2011), which were used to explore the geometry of the whorls along the predatory path.…”

Section: Literature Reviewmentioning

confidence: 99%

Association between shell morphology of micro-land snails (genus Plectostoma) and their predator’s predatory behaviour

Liew

2013

Preprint

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Association between shell morphology of micro-land snails (genus Plectostoma) and their predator's predatory behaviourPredator-prey interactions are among the main ecological interactions that shape the diversity of biological form. In many cases, the evolution of the mollusc shell form is presumably driven by predation. However, the adaptive significance of several uncommon, yet striking, shell traits of land snails are still poorly known. These include the distorted coiled "tuba" and the protruded radial ribs that can be found in micro-landsnails of the genus Plectostoma. Here, we experimentally tested whether these shell traits may act as defensive adaptations against predators. First, we identified the predators, namely, Atopos slugs and Pteroptyx beetle larvae, and their predatory strategies towards Plectostoma snails. Then, we characterised and quantified the possible anti-predation behaviour and shell traits of Plectostoma snails both in terms of their properties and efficiencies in defending against the Atopos slug predatory strategies, namely, shell-apertural entry and shell-drilling. The results showed that Atopos slugs would first attack the snail by shell-apertural entry, and, should this fail, shift to the energetically more costly shell-drilling strategy. We found that the shell tuba of Plectostoma snails is an effective defensive trait against shell-apertural entry attack. None of the snail traits, such as resting behaviour, shell thickness, shell tuba shape, shell rib density and intensity can protect the snail from the slug's shell-drilling attack. However, these traits could increase the predation costs to the slug. Further analysis on the shell traits revealed that the lack of effectiveness these anti-predation shell traits may be caused by a functional trade-off between shell traits under selection of two different predatory strategies. Lastly, we discuss our results in the framework of Red Queen predator-prey coevolution and escalation, and propose several key elements for future study.PeerJ PrePrints | https://peerj.com/preprints/86v1/ | v1 received:

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The Natural 3D Spiral

Cited by 49 publications

References 29 publications

Warp-based helical implicit primitives

Warp-based helical implicit primitives

A method for quantifying, visualising, and analysing gastropod shell form

Association between shell morphology of micro-land snails (genus Plectostoma) and their predator’s predatory behaviour

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