scholarly journals Classifying toric surface codes of dimension 7

2021 ◽  
Vol 14 (4) ◽  
pp. 605-616
Author(s):  
Emily Cairncross ◽  
Stephanie Ford ◽  
Eli Garcia ◽  
Kelly Jabbusch
Keyword(s):  
Toric Surface

Tool path optimal design for slow tool servo turning of complex optical surface

2016 ◽  
Vol 231 (5) ◽  
pp. 825-837 ◽  
Author(s):  
Xu Chen ◽  
Min Kang ◽  
Xingsheng Wang ◽  
Muhammad Hassan ◽  
Jun Yang
Keyword(s):  
Optimal Design ◽  
Tool Path ◽  
Tool Path Generation ◽  
Interpolation Error ◽  
Machining Accuracy ◽  
Calculation Error ◽  
Path Generation ◽  
Optical Surface ◽  
Toric Surface ◽  
Slow Tool Servo

In order to increase the machining accuracy of slow tool servo turning of complex optical surface, the optimal design for tool path was studied. A comprehensive tool path generation strategy was proposed to optimize the tool path for machining complex surfaces. A new algorithm was designed for tool nose radius compensation which had less calculation error. Hermite segment interpolation was analyzed based on integrated multi-axes controller, and a new interpolation method referred to as triangle rotary method was put forward and was compared with the area method and three-point method. The machining simulation indicated that the triangle rotary method was significant in error reduction. The interpolation error of toric surface was reduced to 0.0015 µm from 0.06 µm and sinusoidal array surface’s interpolation error decreases to 0.37 µm from 1.5 µm. Finally, a toric surface was machined using optimum tool path generation method to evaluate the proposed tool path generation method.

scholarly journals Exceptional sequences of invertible sheaves on rational surfaces

2011 ◽  
Vol 147 (4) ◽  
pp. 1230-1280 ◽  
Author(s):  
Lutz Hille ◽  
Markus Perling
Keyword(s):  
Large Class ◽  
Complete Classification ◽  
Toric Surface ◽  
Structural Result ◽  
Toric Surfaces ◽  
Rational Surfaces ◽  
Exceptional Sequences

AbstractIn this article we consider exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that to every such sequence one can associate a smooth complete toric surface in a canonical way. We use this structural result to prove various theorems on exceptional and strongly exceptional sequences of invertible sheaves on rational surfaces. We construct full strongly exceptional sequences for a large class of rational surfaces. For the case of toric surfaces we give a complete classification of full strongly exceptional sequences of invertible sheaves.

scholarly journals MINIMALITY AND MUTATION-EQUIVALENCE OF POLYGONS

2017 ◽  
Vol 5 ◽  
Author(s):  
ALEXANDER KASPRZYK ◽  
BENJAMIN NILL ◽  
THOMAS PRINCE
Keyword(s):  
Mirror Symmetry ◽  
Smooth Surface ◽  
Equivalence Classes ◽  
Del Pezzo Surfaces ◽  
Toric Surface ◽  
Del Pezzo

We introduce a concept of minimality for Fano polygons. We show that, up to mutation, there are only finitely many Fano polygons with given singularity content, and give an algorithm to determine representatives for all mutation-equivalence classes of such polygons. This is a key step in a program to classify orbifold del Pezzo surfaces using mirror symmetry. As an application, we classify all Fano polygons such that the corresponding toric surface is qG-deformation-equivalent to either (i) a smooth surface; or (ii) a surface with only singularities of type$1/3(1,1)$.

scholarly journals Linear Precision for Toric Surface Patches

2009 ◽  
Vol 10 (1) ◽  
pp. 37-66 ◽  
Author(s):  
Hans-Christian Graf von Bothmer ◽  
Kristian Ranestad ◽  
Frank Sottile
Keyword(s):  
Toric Surface ◽  
Surface Patches

scholarly journals A MINIMAL SET OF GENERATORS FOR THE CANONICAL IDEAL OF A NONDEGENERATE CURVE

2014 ◽  
Vol 98 (3) ◽  
pp. 311-323 ◽  
Author(s):  
WOUTER CASTRYCK ◽  
FILIP COOLS
Keyword(s):  
Laurent Polynomial ◽  
Projective Curve ◽  
Toric Surface ◽  
Minimal Set ◽  
Smooth Projective Curve ◽  
Canonical Ideal

We give an explicit way of writing down a minimal set of generators for the canonical ideal of a nondegenerate curve, or of a more general smooth projective curve in a toric surface, in terms of its defining Laurent polynomial.

scholarly journals Toric systems and mirror symmetry

2013 ◽  
Vol 149 (11) ◽  
pp. 1839-1855 ◽  
Author(s):  
Raf Bocklandt
Keyword(s):  
Mirror Symmetry ◽  
Line Bundles ◽  
Del Pezzo Surfaces ◽  
Toric Surface ◽  
Exceptional Sequence ◽  
Del Pezzo Surface ◽  
Rational Surfaces ◽  
Exceptional Sequences ◽  
Del Pezzo ◽  
Compositio Math

AbstractIn their paper [Exceptional sequences of invertible sheaves on rational surfaces, Compositio Math. 147 (2011), 1230–1280], Hille and Perling associate to every cyclic full strongly exceptional sequence of line bundles on a toric weak del Pezzo surface a toric system, which defines a new toric surface. We interpret this construction as an instance of mirror symmetry and extend it to a duality on the set of toric weak del Pezzo surfaces equipped with a cyclic full strongly exceptional sequence.

On classification of toric surface codes of dimension seven

2020 ◽  
Vol 28 (2) ◽  
pp. 263-319
Author(s):  
Naveed Hussain ◽  
Xue Luo ◽  
Stephen S.-T. Yau ◽  
Mingyi Zhang ◽  
Huaiqing Zuo
Keyword(s):  
Toric Surface

Curvature continuity conditions between adjacent toric surface patches

2018 ◽  
Vol 37 (7) ◽  
pp. 469-477
Author(s):  
Lanyin Sun ◽  
Chungang Zhu
Keyword(s):  
Toric Surface ◽  
Surface Patches ◽  
Curvature Continuity

Freeform machining of ophthalmic toric lens mould using fast tool servo-assisted ultra-precision diamond turning process

2020 ◽  
pp. 251659842093974
Author(s):  
Ishan Anand Singh ◽  
Gopi Krishna S. ◽  
T. Narendra Reddy ◽  
Prakash Vinod
Keyword(s):  
Surface Roughness ◽  
Single Point ◽  
Measurement Data ◽  
Surface Profile ◽  
Diamond Turning ◽  
Fast Tool Servo ◽  
Toric Surface ◽  
Machined Surface ◽  
Ultra Precision ◽  
A Minor

This research aims to establish a methodology for machining of toric lenses, using fast tool servo-assisted single point diamond turning and to assess the generated surface for its characteristics. Using the established mathematical model, toric surface is explained to understand the geometry and to generate the parameters required for fast tool servo machining. A toric surface with a major diameter of 18.93 mm and a minor diameter of 15.12 mm has been cut on the intelligent ultra-precision turning machine (iUPTM). The surface profile and surface roughness were measured. After analysing the measurement data of the machined surface, on two perpendicular axes of the toric lens, form accuracy of 0.49 µm peak-to-valley (PV), and surface roughness of 12 nm in Ra, 4–8 nm in Sa are obtained. From the experimental results obtained, it can be concluded that the proposed method is a reasonable alternative for manufacturing toric lens mould.

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