Although the act of cutting a single potato (Solanum tuberosum) into french fries may appear to be trivial, the questions concerning the efficiency of this process on an industrial scale are quite daunting. Therefore, many producers are looking for a rigorous method to evaluate the market potential of a given potato crop by predicting the number and parameters of the fries that can be cut from it. Applying the methods of geometry and numerical analysis our group was able to propose several algorithms that can be directly incorporated into the existing production process. Keywords: French fries, geometry, cutting, Finite Fry Method, simulations, histogram

Banagaaya, N.

Budko, N.

Doorn, van, H.

Khimshiashvili, G.

Klooster, Rob

Lindenbergh, P.J.

Verdijck, J.

Vermolen, Fred

Zwaan, I.N.

English

Although the act of cutting a single potato (Solanum tuberosum) into french fries may appear to be trivial, the questions concerning the efficiency of this process on an industrial scale are quite daunting. Therefore, many producers are looking for a rigorous method to evaluate the market potential of a given potato crop by predicting the number and parameters of the fries that can be cut from it. Applying the methods of geometry and numerical analysis our group was able to propose several algorithms that can be directly incorporated into the existing production process.
Keywords: French fries, geometry, cutting, Finite Fry Method, simulations, histogram

Banagaaya, N Nico

Budko, Neil

Doorn, H  van

Khimshiashvili, G

Lindenbergh, PJ

Verdijck, J

Zwaan, IN Ian

Repository TU/e

The mathematics of french fries

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 The mathematics of french friesCitation for published version (APA):Banagaaya, N., Budko, N., Doorn, van, H., Khimshiashvili, G., Klooster, R., Lindenbergh, P. J., Verdijck, J.,Vermolen, F., & Zwaan, I. N. (2014). The mathematics of french fries. In J. Dubbeldam, W. Groenevelt, A. W.Heemink, D. Lahaye, C. Meerman, & F. Meulen, van der (Eds.), Proceedings of the Study Group Mathematicswith Industry (Delft, The Netherlands, January 27-31, 2014) (pp. 24-33). Technische Universiteit Delft.Document status and date:Published: 01/01/2014Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 05. Oct. 2023ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 24 — #30 iiiiiiThe Mathematics of French FriesNicodemus Banagaaya1,Neil Budko2 ∗,Hans van Doorn3†,Giorgi Khimshiashvili4,Rob Klooster3,Pieter-Jelte Lindenbergh3,Jacqueline Verdijck3,Fred Vermolen2and Ian Zwaan11 TU Eindhoven2 TU Delft3 HZPC4 Ilia State Univ., GeorgiaAbstractAlthough the act of cutting a single potato (Solanum tuberosum)into french fries may appear to be trivial, the questions concerningthe efficiency of this process on an industrial scale are quite daunt-ing. Therefore, many producers are looking for a rigorous method toevaluate the market potential of a given potato crop by predicting thenumber and parameters of the fries that can be cut from it. Applyingthe methods of geometry and numerical analysis our group was able topropose several algorithms that can be directly incorporated into theexisting production process.Keywords: French fries, geometry, cutting, Finite Fry Method, simu-lations, histograms∗For questions about mathematics: n.v.budko@tudelft.nl†For questions about potatoes: hans.vandoorn@hzpc.nlProceedings of the SWI 2014 Held in Delft24ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 25 — #31 iiiiii1 IntroductionThe HZPC Holland B.V. company is a major international supplier of pota-toes. The question posed by HZPC could be summarized as follows:Question: Given a set of potato tubers with approximate informationabout the size of each tuber and the desired cross-section of a fry, how manyfrench fries of various lengths and textural quality can be obtained from thisset, and what will be the volume fraction of waste?During the SWI week our group was able to solve the geometrical part ofthe problem. Whereas, the question concerning the textural quality remainspartially open. Below we describe two methods to cut tubers into virtualfries and estimate their number, their length distribution, and the amount ofwaste. The analytical approach provides with the formula for the volume ofeach tuber based on the best elliptic approximation of the surface and with aquick and dirty way to compute the fry length histogram from experimentaldata. The second approach, which we call the Finite Fry Method (FFM), hasbeen implemented as two separate numerical algorithms: one that is tunedto work with the currently available experimental data, and the other thatallows generating tubers of arbitrary shape and perform Monte-Carlo studieson the obtained ensemble. Finally, we have developed an analytical procedureto interpolate the DMC quality trait distribution sampled in a small numberof points onto the whole volume of the tuber.2 Geometrical modelingThe available information about the shape of each tuber consists of its maxi-mal extent (length) and several measurements in other directions. Namely, innine equidistant planes orthogonal to the length direction the measurementsof the width and height are available, two in each plane. However, although itis known that the directions of the width and height are mutually orthogonaland consistent over all nine planes, the measurements do not provide with theactual coordinates of the boundary points.This limited information allows constructing only a simplified geometricalmodel of a tuber. We consider the intersection of the tuber boundary surfacewith each of the nine cross-sectional measurement planes to be a concentricbut not necessarily a confocal ellipse with known semi-axes.Computing the volume of a tuber requires interpolating its boundary sur-face between the aforementioned cross-sectional ellipses. Depending on thechosen interpolation model the surface between the planes may or may nothave an elliptic cross-section. Without losing much precision we assume thatall cross-sections of the tuber in the direction orthogonal to its length are in-deed elliptic. The advantage of this assumption is the ability to compute theProceedings of the SWI 2014 Held in Delft25ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 26 — #32 iiiiii4000 4050 4100 4150 4200 4250101102103tuber number  volume from weight and densityvolume estimated from geometryFigure 1: Calculated volume of the geometric model compared with the vol-ume estimated from the weight and density of tubers.volume of a tuber explicitly as a sum of the volumes of eight twisted ellipticcylinders defined by the measurement planes plus the volume of two caps thatcan be modeled as elliptic cones. The result of computing the volume via thisapproach is shown in Figure 1 where it is compared with the volume estimatebased on the weight and density of a tuber.Further improvement of this geometric model may consist in a better de-scription of the tuber caps, for example, as elliptic paraboloids instead ofelliptic cones.3 Cutting potatoes into friesWith the model of the tuber boundary surface at hand one can proceed ‘cut-ting’ the potato into fries and estimating the waste – fries that do not conformto the industry standards.Let the z-axis be along the length direction. Introduce a uniform two-dimensional Cartesian grid in the horizontal (x, y)-plane with the grid steph – the chosen width of a fry (h = 6, 7, 8, 9, 10, 12, 14, or 16 mm). Cut thetuber into fries numerically by constructing finite elements that extend alongthe z-axis with their vertical edges coinciding with the grid points of thepreviously defined horizontal grid. The surface of the tuber is approximatedProceedings of the SWI 2014 Held in Delft26ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 27 — #33 iiiiii−20020−2002010203040506070−20020−200202030405060708090100−20020−200202030405060708090100110Figure 2: Three tubers from the 40 − 50 mm class (experimental data) cutinto fries of width h = 6 mm. Only acceptable fries are indicated.in a piecewise linear fashion at this stage. Discard all fries that do not conformto the industry standard and compute the waste.The result of the application of this numerical algorithm, which we callthe Finite Fry Method (FFM), to the first three tubers in the 40 − 50 mmwidth class is shown in Figure 2.Although, we were able to run this algorithm on as many as 11333 tubers,the discretization of a large number of tubers into fries may take significanttime (about an hour on a laptop). Therefore, we have also developed anapproximate but simple method to estimate the number of fries that can becut from each tuber. The main idea is to count only the fries that have a squarecross-section discarding all wedge-shaped fries that are cut from the sides of atuber. This method detects the smallest of the cross-sectional ellipses of eachtuber, and is extremely fast (seconds) and surprisingly accurate (see Figure 3).4 Modelling variations in tuber geometryAnother realization of the finite-fry method (FFM) was developed to relax thecurrent experimentally imposed limitations on the tuber geometry. Namely,as was mentioned above, the measurements did not provide with the actualcoordinates of the boundary points and forced us to assume the concentricelliptical geometry when working with the data. In reality, however, tuberscan have very asymmetric shapes. In this second realization of the FFMwe have implemented a more systematic finite-element approach and havesimulated a large statistical ensemble of asymmetric tubers.As before, the potato shape is determined by a number of slices with ellip-tic contours orthogonal to its maximal (length, vertical) dimension. However,Proceedings of the SWI 2014 Held in Delft27ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 28 — #34 iiiiii0 50 100 15000.0050.010.0150.020.0250.030.0350.04length, [mm]  tubersfries, simplefries, finite elements0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90246810121416volume fractionFigure 3: Left: Distribution of lengths of all acceptable 6 mm wide friesthat can be cut out of 11333 tubers from the 40 − 50 mm class calculatedusing a simple geometrical estimate and a more exact numerical procedure.Also shown is the distribution of the tuber length within the class. Right:Distribution of the volume fraction of waste.this time not only the dimensions and orientation of the ellipses are allowedto vary but also their centres are allowed to shift off the length axis. InFigure 4 we show a schematic representation of tubers defined by a set ofalgorithmically generated ellipses.To get the number of fries, we use a background (horizontal) square domainwith the width and height chosen in such a way that all tubers of the simulatedensemble can be projected onto this domain. This domain is referred to asthe reference surface. This area is divided into squares of equal size, which wecall elements. Each elliptic slice is projected onto the reference area so thateach square fits either completely, or partially, or not at all in the projectedslice.To make the whole procedure more systematic we construct an orderedlist containing the indexes of all points in groups of four – the vertices of eachelement. Such lists are usually referred to as the topology of the grid. Thislist makes it easy to determine whether a fry belongs to a projected ellipseby testing for each vertex and each element whether the coordinates (x, y) ofthe vertex satisfy(xa)2+(yb)2< 1.If that is the case, then the vertex point and thus the edge of the fry is insidethe tuber. If an element, which represents the cross-section of a fry, fits eitherProceedings of the SWI 2014 Held in Delft28ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 29 — #35 iiiiii2468456024681012heightwidthlength4 56456024681012141618Figure 4: Two examples (Case I and Case II) of computer-generated tubersdefined by ellipses with variable length, width, and center offset. Here, thelength and width are deterministic functions, whereas the coordinates of thecentres are taken from a normal distribution with zero mean and the standarddeviation of 10 mmpartially or entirely within the projection of the slice onto the reference area,then this element is interpreted as a ‘fry’.Among all the fries we determine the list of accepted fries by accepting afry if the intersection of the corresponding element with the projected ellipsehas an area of at least 0.7A, where A denotes the element area. A schematicof accepted and rejected fries over a single slice is given in Figure 5. Thisprocedure is repeated for all the elliptic slices of the tuber.The length of a fry corresponding to a certain element is determined byadding all the distances between adjacent slices that have yielded an accept-able intersection area with this element during the previous step. Due to theallowed variations in the tuber geometry some fries may have to be cut intopieces. In such cases each piece is considered to be a separate fry. As before,fries are only accepted if their length exceeds 40 mm. All the acceptancecriteria for the fries can be changed in the model whenever necessary.In the Monte-Carlo simulations, the tuber length, the centres, and theorientation and length of the semi-axes of the elliptic cross-sections are allassumed to be random normally distributed variables with adjustable meansand variances. Furthermore, the mean width and height of each ellipse changeaccording to a prescribed function in the length direction. We note that thisfunction can be adjusted to model any desired shape.Similarly to Figure 3 we present the results by comparing the distributionof the length of tubers (which is an input parameter) to the distribution of theProceedings of the SWI 2014 Held in Delft29ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 30 — #36 iiiiii0 1 2 3 4 5 6 7 8 9 10012345678910x−position (cm)y−position (cm)Figure 5: A schematic representation of accepted fries over an elliptic slice ofa tuber. The red and blue dots, respectively, indicate the centres of the friesthat are rejected and acceptedlength of accepted fries (output parameter). As a tuber shape, we considera mean tuber length of L = 150 mm, with the standard deviation of 35 mm,that is L ∼ N (150, 352). The elliptic projections were simulated according tothe formula:(x− xca(z))2+(y − ycb(z))2= 1,where z is the vertical coordinate and xc, yc ∼ N (0, 102) are the center co-ordinates taken from a normal distribution. We present the results for thefollowing two shape functions:E{a(z)} = 4Lz(L− z) = E{b(z)}, Case I; (1)E{a(z)} =(4Lz(L− z)) 18= E{b(z)}, Case II; (2)where E{. . . } denotes the expectation used in the Monte-Carlo simulations.Note that we also set a = b in the current simulations but we have theflexibility to use any value.The results are presented in Figure 6. It can be seen that, in agreementwith the experimental results of Figure 3, the average length of a fry is lowerProceedings of the SWI 2014 Held in Delft30ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 31 — #37 iiiiii0 5 10 15 20 2500.050.10.150.20.250.30.35length (cm)TubersFries0 5 10 15 20 2500.050.10.150.20.250.30.35length (cm)Tuber lengthFry lengthFigure 6: Statistical distribution of the tuber and fry lengths for two classesof tuber shapes, see (1)–(2).than the average tuber length. This is a direct consequence of the largenumber of fries that is cut from the part of the tuber that is further awayfrom the main axis. Along the main axis the length of fries is (approximately)equal to the tuber length, whereas at positions away from the main axis thefries are shorter and more numerous. This causes the relatively large numberof fries with a length of 40 mm in the first case.While the tuber shape is more ellipsoidal in the first case, in the secondcase the shape is more cylindrical (see Figure 4). Therefore, in the secondcase, the number of small fries is smaller. Thus, the proposed mathematicalmodels can be used to quantify the observations. In particular, it is interestingto observe that the tuber geometry determines the shape of the probabilitydensity function.5 Fry texture and the DMC quality traitAnother question posed by the HZPC company was to investigate the textureof fries after frying and the so-called DMC quality trait of raw fries. The DMCtrait is currently measured in just a few points over the volume of some of thetubers. To interpolate the very sparse measurements of the DMC so that theycan be used to determine the DMC of each individual fry we have introducedand computed the distance between an inner point of an elliptic slice and itsbounding ellipse. This distance is then inserted into a parametric expressionapproximating the more recent fine-scale measurements of the DMC:φ(x, y) = φc − α(dist((x, y); Γ))2, (3)Proceedings of the SWI 2014 Held in Delft31ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 32 — #38 iiiiiix−position (cm)y−position (cm)  0 1 2 3 4 5 6 7 8 9 100123456789100.20.220.240.260.280.30.320.340.360.380.4Figure 7: The distribution of the DMC quality trait over an elliptic slice.where (x, y), φc, α and Γ, respectively, represent any point within the ellipse,the DMC content near the tuber surface (the maximum), and two tunableconstants. The spatial distribution of DMC is denoted by φ. The proposedfunction is shown in Figure 7. We note that all tunable parameters can beeasily determined by an optimization algorithm. In the current simulation,φ0 = 0.4, and α = 0.025. We did not move further into this direction duringthe SWI-week.6 ConclusionsThe problem posed by the HZPC company was to find a mathematical tech-nique that could help evaluate the market potential of a given set of potatotubers. For the particular French fries market the main quality factor is theamount and type of fries that can be cut from the tubers. As tubers vary inshape and size even after having been sorted into several size-related classes,the best approach appears to be the analysis of a histogram depicting thedistribution of length among the fries. In order to obtain such a histogramwe have developed an approximate and fast analytical procedure and a moreexact numerical technique, which both cut tubers into virtual fries with achosen cross-section. The advantage of the numerical approach stems fromthe fact that it allows investigating the influence of rather arbitrary shapeperturbations on the histogram by running a Monte-Carlo simulation with acomputer generated set of tubers. Both methods were tested on the providedexperimental data and showed similar results. The analytical expression forProceedings of the SWI 2014 Held in Delft32ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 33 — #39 iiiiiithe volume of each tuber based on the available spatial measurements is ingood agreement with the volume computed from the weight and density data.The second part of the question concerned the distribution of the DMCquality trait inside the tubers and its relation to the texture of French fries.In this respect we have proposed a parametric expression that can be fitto the available sparse experimental data in order to interpolate the DMCdensity over the fry volume. Armed with this DMC trait distribution onecould extend the histogram approach by showing the statistics of the DMCfor each fry length.Proceedings of the SWI 2014 Held in Delft33

 The mathematics of french friesCitation for published version (APA):Banagaaya, N., Budko, N., Doorn, van, H., Khimshiashvili, G., Klooster, R., Lindenbergh, P. J., Verdijck, J.,Vermolen, F., & Zwaan, I. N. (2014). The mathematics of french fries. In J. Dubbeldam, W. Groenevelt, A. W.Heemink, D. Lahaye, C. Meerman, & F. Meulen, van der (Eds.), Proceedings of the Study Group Mathematicswith Industry (Delft, The Netherlands, January 27-31, 2014) (pp. 24-33). Technische Universiteit Delft.Document status and date:Published: 01/01/2014Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 16. Nov. 2023ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 24 — #30 iiiiiiThe Mathematics of French FriesNicodemus Banagaaya1,Neil Budko2 ∗,Hans van Doorn3†,Giorgi Khimshiashvili4,Rob Klooster3,Pieter-Jelte Lindenbergh3,Jacqueline Verdijck3,Fred Vermolen2and Ian Zwaan11 TU Eindhoven2 TU Delft3 HZPC4 Ilia State Univ., GeorgiaAbstractAlthough the act of cutting a single potato (Solanum tuberosum)into french fries may appear to be trivial, the questions concerningthe efficiency of this process on an industrial scale are quite daunt-ing. Therefore, many producers are looking for a rigorous method toevaluate the market potential of a given potato crop by predicting thenumber and parameters of the fries that can be cut from it. Applyingthe methods of geometry and numerical analysis our group was able topropose several algorithms that can be directly incorporated into theexisting production process.Keywords: French fries, geometry, cutting, Finite Fry Method, simu-lations, histograms∗For questions about mathematics: n.v.budko@tudelft.nl†For questions about potatoes: hans.vandoorn@hzpc.nlProceedings of the SWI 2014 Held in Delft24ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 25 — #31 iiiiii1 IntroductionThe HZPC Holland B.V. company is a major international supplier of pota-toes. The question posed by HZPC could be summarized as follows:Question: Given a set of potato tubers with approximate informationabout the size of each tuber and the desired cross-section of a fry, how manyfrench fries of various lengths and textural quality can be obtained from thisset, and what will be the volume fraction of waste?During the SWI week our group was able to solve the geometrical part ofthe problem. Whereas, the question concerning the textural quality remainspartially open. Below we describe two methods to cut tubers into virtualfries and estimate their number, their length distribution, and the amount ofwaste. The analytical approach provides with the formula for the volume ofeach tuber based on the best elliptic approximation of the surface and with aquick and dirty way to compute the fry length histogram from experimentaldata. The second approach, which we call the Finite Fry Method (FFM), hasbeen implemented as two separate numerical algorithms: one that is tunedto work with the currently available experimental data, and the other thatallows generating tubers of arbitrary shape and perform Monte-Carlo studieson the obtained ensemble. Finally, we have developed an analytical procedureto interpolate the DMC quality trait distribution sampled in a small numberof points onto the whole volume of the tuber.2 Geometrical modelingThe available information about the shape of each tuber consists of its maxi-mal extent (length) and several measurements in other directions. Namely, innine equidistant planes orthogonal to the length direction the measurementsof the width and height are available, two in each plane. However, although itis known that the directions of the width and height are mutually orthogonaland consistent over all nine planes, the measurements do not provide with theactual coordinates of the boundary points.This limited information allows constructing only a simplified geometricalmodel of a tuber. We consider the intersection of the tuber boundary surfacewith each of the nine cross-sectional measurement planes to be a concentricbut not necessarily a confocal ellipse with known semi-axes.Computing the volume of a tuber requires interpolating its boundary sur-face between the aforementioned cross-sectional ellipses. Depending on thechosen interpolation model the surface between the planes may or may nothave an elliptic cross-section. Without losing much precision we assume thatall cross-sections of the tuber in the direction orthogonal to its length are in-deed elliptic. The advantage of this assumption is the ability to compute theProceedings of the SWI 2014 Held in Delft25ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 26 — #32 iiiiii4000 4050 4100 4150 4200 4250101102103tuber number  volume from weight and densityvolume estimated from geometryFigure 1: Calculated volume of the geometric model compared with the vol-ume estimated from the weight and density of tubers.volume of a tuber explicitly as a sum of the volumes of eight twisted ellipticcylinders defined by the measurement planes plus the volume of two caps thatcan be modeled as elliptic cones. The result of computing the volume via thisapproach is shown in Figure 1 where it is compared with the volume estimatebased on the weight and density of a tuber.Further improvement of this geometric model may consist in a better de-scription of the tuber caps, for example, as elliptic paraboloids instead ofelliptic cones.3 Cutting potatoes into friesWith the model of the tuber boundary surface at hand one can proceed ‘cut-ting’ the potato into fries and estimating the waste – fries that do not conformto the industry standards.Let the z-axis be along the length direction. Introduce a uniform two-dimensional Cartesian grid in the horizontal (x, y)-plane with the grid steph – the chosen width of a fry (h = 6, 7, 8, 9, 10, 12, 14, or 16 mm). Cut thetuber into fries numerically by constructing finite elements that extend alongthe z-axis with their vertical edges coinciding with the grid points of thepreviously defined horizontal grid. The surface of the tuber is approximatedProceedings of the SWI 2014 Held in Delft26ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 27 — #33 iiiiii−20020−2002010203040506070−20020−200202030405060708090100−20020−200202030405060708090100110Figure 2: Three tubers from the 40 − 50 mm class (experimental data) cutinto fries of width h = 6 mm. Only acceptable fries are indicated.in a piecewise linear fashion at this stage. Discard all fries that do not conformto the industry standard and compute the waste.The result of the application of this numerical algorithm, which we callthe Finite Fry Method (FFM), to the first three tubers in the 40 − 50 mmwidth class is shown in Figure 2.Although, we were able to run this algorithm on as many as 11333 tubers,the discretization of a large number of tubers into fries may take significanttime (about an hour on a laptop). Therefore, we have also developed anapproximate but simple method to estimate the number of fries that can becut from each tuber. The main idea is to count only the fries that have a squarecross-section discarding all wedge-shaped fries that are cut from the sides of atuber. This method detects the smallest of the cross-sectional ellipses of eachtuber, and is extremely fast (seconds) and surprisingly accurate (see Figure 3).4 Modelling variations in tuber geometryAnother realization of the finite-fry method (FFM) was developed to relax thecurrent experimentally imposed limitations on the tuber geometry. Namely,as was mentioned above, the measurements did not provide with the actualcoordinates of the boundary points and forced us to assume the concentricelliptical geometry when working with the data. In reality, however, tuberscan have very asymmetric shapes. In this second realization of the FFMwe have implemented a more systematic finite-element approach and havesimulated a large statistical ensemble of asymmetric tubers.As before, the potato shape is determined by a number of slices with ellip-tic contours orthogonal to its maximal (length, vertical) dimension. However,Proceedings of the SWI 2014 Held in Delft27ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 28 — #34 iiiiii0 50 100 15000.0050.010.0150.020.0250.030.0350.04length, [mm]  tubersfries, simplefries, finite elements0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90246810121416volume fractionFigure 3: Left: Distribution of lengths of all acceptable 6 mm wide friesthat can be cut out of 11333 tubers from the 40 − 50 mm class calculatedusing a simple geometrical estimate and a more exact numerical procedure.Also shown is the distribution of the tuber length within the class. Right:Distribution of the volume fraction of waste.this time not only the dimensions and orientation of the ellipses are allowedto vary but also their centres are allowed to shift off the length axis. InFigure 4 we show a schematic representation of tubers defined by a set ofalgorithmically generated ellipses.To get the number of fries, we use a background (horizontal) square domainwith the width and height chosen in such a way that all tubers of the simulatedensemble can be projected onto this domain. This domain is referred to asthe reference surface. This area is divided into squares of equal size, which wecall elements. Each elliptic slice is projected onto the reference area so thateach square fits either completely, or partially, or not at all in the projectedslice.To make the whole procedure more systematic we construct an orderedlist containing the indexes of all points in groups of four – the vertices of eachelement. Such lists are usually referred to as the topology of the grid. Thislist makes it easy to determine whether a fry belongs to a projected ellipseby testing for each vertex and each element whether the coordinates (x, y) ofthe vertex satisfy(xa)2+(yb)2< 1.If that is the case, then the vertex point and thus the edge of the fry is insidethe tuber. If an element, which represents the cross-section of a fry, fits eitherProceedings of the SWI 2014 Held in Delft28ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 29 — #35 iiiiii2468456024681012heightwidthlength4 56456024681012141618Figure 4: Two examples (Case I and Case II) of computer-generated tubersdefined by ellipses with variable length, width, and center offset. Here, thelength and width are deterministic functions, whereas the coordinates of thecentres are taken from a normal distribution with zero mean and the standarddeviation of 10 mmpartially or entirely within the projection of the slice onto the reference area,then this element is interpreted as a ‘fry’.Among all the fries we determine the list of accepted fries by accepting afry if the intersection of the corresponding element with the projected ellipsehas an area of at least 0.7A, where A denotes the element area. A schematicof accepted and rejected fries over a single slice is given in Figure 5. Thisprocedure is repeated for all the elliptic slices of the tuber.The length of a fry corresponding to a certain element is determined byadding all the distances between adjacent slices that have yielded an accept-able intersection area with this element during the previous step. Due to theallowed variations in the tuber geometry some fries may have to be cut intopieces. In such cases each piece is considered to be a separate fry. As before,fries are only accepted if their length exceeds 40 mm. All the acceptancecriteria for the fries can be changed in the model whenever necessary.In the Monte-Carlo simulations, the tuber length, the centres, and theorientation and length of the semi-axes of the elliptic cross-sections are allassumed to be random normally distributed variables with adjustable meansand variances. Furthermore, the mean width and height of each ellipse changeaccording to a prescribed function in the length direction. We note that thisfunction can be adjusted to model any desired shape.Similarly to Figure 3 we present the results by comparing the distributionof the length of tubers (which is an input parameter) to the distribution of theProceedings of the SWI 2014 Held in Delft29ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 30 — #36 iiiiii0 1 2 3 4 5 6 7 8 9 10012345678910x−position (cm)y−position (cm)Figure 5: A schematic representation of accepted fries over an elliptic slice ofa tuber. The red and blue dots, respectively, indicate the centres of the friesthat are rejected and acceptedlength of accepted fries (output parameter). As a tuber shape, we considera mean tuber length of L = 150 mm, with the standard deviation of 35 mm,that is L ∼ N (150, 352). The elliptic projections were simulated according tothe formula:(x− xca(z))2+(y − ycb(z))2= 1,where z is the vertical coordinate and xc, yc ∼ N (0, 102) are the center co-ordinates taken from a normal distribution. We present the results for thefollowing two shape functions:E{a(z)} = 4Lz(L− z) = E{b(z)}, Case I; (1)E{a(z)} =(4Lz(L− z)) 18= E{b(z)}, Case II; (2)where E{. . . } denotes the expectation used in the Monte-Carlo simulations.Note that we also set a = b in the current simulations but we have theflexibility to use any value.The results are presented in Figure 6. It can be seen that, in agreementwith the experimental results of Figure 3, the average length of a fry is lowerProceedings of the SWI 2014 Held in Delft30ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 31 — #37 iiiiii0 5 10 15 20 2500.050.10.150.20.250.30.35length (cm)TubersFries0 5 10 15 20 2500.050.10.150.20.250.30.35length (cm)Tuber lengthFry lengthFigure 6: Statistical distribution of the tuber and fry lengths for two classesof tuber shapes, see (1)–(2).than the average tuber length. This is a direct consequence of the largenumber of fries that is cut from the part of the tuber that is further awayfrom the main axis. Along the main axis the length of fries is (approximately)equal to the tuber length, whereas at positions away from the main axis thefries are shorter and more numerous. This causes the relatively large numberof fries with a length of 40 mm in the first case.While the tuber shape is more ellipsoidal in the first case, in the secondcase the shape is more cylindrical (see Figure 4). Therefore, in the secondcase, the number of small fries is smaller. Thus, the proposed mathematicalmodels can be used to quantify the observations. In particular, it is interestingto observe that the tuber geometry determines the shape of the probabilitydensity function.5 Fry texture and the DMC quality traitAnother question posed by the HZPC company was to investigate the textureof fries after frying and the so-called DMC quality trait of raw fries. The DMCtrait is currently measured in just a few points over the volume of some of thetubers. To interpolate the very sparse measurements of the DMC so that theycan be used to determine the DMC of each individual fry we have introducedand computed the distance between an inner point of an elliptic slice and itsbounding ellipse. This distance is then inserted into a parametric expressionapproximating the more recent fine-scale measurements of the DMC:φ(x, y) = φc − α(dist((x, y); Γ))2, (3)Proceedings of the SWI 2014 Held in Delft31ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 32 — #38 iiiiiix−position (cm)y−position (cm)  0 1 2 3 4 5 6 7 8 9 100123456789100.20.220.240.260.280.30.320.340.360.380.4Figure 7: The distribution of the DMC quality trait over an elliptic slice.where (x, y), φc, α and Γ, respectively, represent any point within the ellipse,the DMC content near the tuber surface (the maximum), and two tunableconstants. The spatial distribution of DMC is denoted by φ. The proposedfunction is shown in Figure 7. We note that all tunable parameters can beeasily determined by an optimization algorithm. In the current simulation,φ0 = 0.4, and α = 0.025. We did not move further into this direction duringthe SWI-week.6 ConclusionsThe problem posed by the HZPC company was to find a mathematical tech-nique that could help evaluate the market potential of a given set of potatotubers. For the particular French fries market the main quality factor is theamount and type of fries that can be cut from the tubers. As tubers vary inshape and size even after having been sorted into several size-related classes,the best approach appears to be the analysis of a histogram depicting thedistribution of length among the fries. In order to obtain such a histogramwe have developed an approximate and fast analytical procedure and a moreexact numerical technique, which both cut tubers into virtual fries with achosen cross-section. The advantage of the numerical approach stems fromthe fact that it allows investigating the influence of rather arbitrary shapeperturbations on the histogram by running a Monte-Carlo simulation with acomputer generated set of tubers. Both methods were tested on the providedexperimental data and showed similar results. The analytical expression forProceedings of the SWI 2014 Held in Delft32ii“swi-2014-delft-proceedings” — 2014/4/22 — 11:31 — page 33 — #39 iiiiiithe volume of each tuber based on the available spatial measurements is ingood agreement with the volume computed from the weight and density data.The second part of the question concerned the distribution of the DMCquality trait inside the tubers and its relation to the texture of French fries.In this respect we have proposed a parametric expression that can be fitto the available sparse experimental data in order to interpolate the DMCdensity over the fry volume. Armed with this DMC trait distribution onecould extend the histogram approach by showing the statistics of the DMCfor each fry length.Proceedings of the SWI 2014 Held in Delft33

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The mathematics of french fries

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