At first, his experience is a heady rush of expectation and optimism. He finds other Chinese students to share a cheap apartment with him, and flirts with an attractive American girl who works in a local tea shop. When the head of the department, Jacob Reiser Aidan Quinn , welcomes Liu Xing into his select cosmology group, it seems that only hard work stands between him and a bright future in American science. At an orientation for foreigners sponsored by a local church, Joanna Silver Meryl Streep , a wealthy patron of the university, notices the earnest student.
An unspoken bond forms between them. Liu Xing becomes Reisers protg, accompanying him to a prestigious conference where he makes an impressive debut. He is drawn to the study of dark matter, an unseen substance that shapes the universe, but it soon becomes clear that his developing theories threaten Reisers conflicting theories and well-established studies. Excited by the possibility of a breakthrough, Liu Xing is deaf to warnings that he must first pay his dues.
When he is eclipsed within the department by Laurence, a more dutiful Chinese student, Liu Xing is forced to go behind Reisers back to publish his discoveries. When the article draws ire instead of accolades, he turns to Joanna, who naively encourages him on his collision course.
Liu Xing clings to the idea of American science as a free market of ideas, and American society as wide open to immigrants. But in the end, his dissertation is rejected, and the girl in the tea shop brushes him off. His roommates find jobs, leaving him behind. Too proud to accept help from Joanna, and unwilling to return home to his parents, Liu Xing becomes a ghost-like presence at the university.
Left alone with his shattered dreams, he explodes in a final act of violence. Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support?
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The ad does not inform my purchase. The video does not play. There is too much buffering. The audio is poor or missing. Video is unrelated to the product. Please fill out the copyright form to register a complaint. Share your thoughts with other customers. Apart for one or two notable exceptions, i historical developments are presented as an evolution of mathematical equations, without much commentary as to their wider importance for humankind.
These texts assume a sophisticated knowledge of advanced mathematics, which, sadly, this author does not possess. Thus, this effort to unravel the story of the spline may be the first nontechnical account written for the intuitive understanding and benefit of the user base — ie. Demystifying the curve, I hope, will help to develop a better appreciation — perhaps even a connoisseurship — of what has now become a de facto architectural tool: Evidence from Mesolithic settlements, as well as vernacular structures like the native American wigwam, show one of the simplest forms of shelter, formed by ramming poles into a circular formation, bending them inward, and lashing them together to form a sturdy dome-shaped latticework.
By the Mesolithic era, humans had learnt to fashion bows in order to hunt game. As a hunter draws the bow, its two tips are pulled closer, producing greater curvature. As the timber is deformed, it stores elastic potential energy as the wood tries to regain its original straightness. When a length of wood is flexed, the material on the concave side goes into compression while the convex side is under tension. In bending, these forces distribute themselves as evenly as possible throughout the length of wood, producing a optimally smooth curve.
Evidence of Mesolithic settlements, as well as numerous indigenous cultures, show primitive shelters were constructed in a similar fashion. The tops of opposite poles were lashed together to form an overall dome-like latticework and covered by animal hides, bark or grass. The native american wigwam is a more recent example. The elasticity of each timber pole produces an arch — an optimal structural shape in compression, imparting strength and rigidity to the crude dome see Figure 1.
Cross sections such as these recorded vessels on contractual order. The outer member marked Fig. Portions of each cross section are indexed to corresponding pieces on the reconfigurable mould.
William Sutherland, The Shipbuilders Assistant: Since ancient history, humankind has harnessed the inherent utility of curved objects, not only to satisfy his basic needs for food and shelter, but also for transportation. In a crude sense, a boat is the simple inversion of the aforementioned primitive hut. When the skin or other outer covering is made watertight with tar, one has a basic albeit unstable dingy. The design and construction of boats has been subsequently refined over millennia, streamlined by the dynamics of water, wind, and speed.
When covered with planking, these skeletal ribs underpin the smooth contours of the hull. The curved profile of each rib was traced from a reconfigurable master template, consisting of piecewise arced and straight wooden pieces. Graduated marks on the template indicated where the pieces were to be repositioned for each successive rib. Using this early parametric method, a fleet of identical ships could be manufactured from a single template. Edson International with kind permission.
The practice was refined throughout the northern Mediterranean well into the s. The practice of plotting the patterns for ribs and keels became known as lofting since the attic above the workshop was the only dry unobstructed floorspace large enough to accommodate the 1: The long curves were scribed onto the timber from a thin flexible strip of timber or steel, called a spline see Figure 3.
The spline is bent and held in place on a flat surface by a series of three or more hooked metal weights, called ducks.
An optimally smooth, attractive, and mechanically sound curvature is guaranteed by the uniform distribution of stress throughout the long elastic spline as it tries to regain its original straightness. By the s, European shipbuilders had begun to rely on smaller scaled drawings for design and contractual documentation.
However, when translating drawings into full-scale parts, boatbuilders still employ the traditional lofting technique using a spline to this day. Set of French Curves devised by the German kinematician and geometer Ludwig Burmester, as illustrated in the Lexikon der gesamten Technik dictionary of technology from by Otto Lueger. The great Italian Renaissance architect Andrea Palladio describes in his famous treatise, The Four Books of Architecture, how he used a flexible ruler to trace the graceful swelling profile of classical columns.
The German kinematician and geometer Ludwig Burmester published a standardized set of curve templates see Figure 4 around Lueger, Each scroll-like template is an amalgamation of assorted mathematical curves.
To use it, a draftsperson finds a close fit along the edge of the tool and repositions it to continue drawing tangential curves. Until the advent of modern computing, French Curves were the tool of choice for architects when drawing any irregular smooth object.
Within the first 15 minutes of the show, orders were received. The DS also had artistic leanings thanks to the hands of Italian sculptor Flaminio Bertoni, who collaborated in shaping its exotic form. The DS was not merely a product of streamlining. It was also the first mass-produced unibody automobile, meaning that the rounded metal skin supported structural loads rather than relying on any internal steel frame. More than fifty years after its launch, the DS is still idolized particularly by designers for its idiosyncratic functionality resolved within an elegant form.
Barthes summarized this poetic integration quite succinctly:. It is well known that smoothness is always an attribute of perfection because its opposite reveals a technical and typically human operation of assembling. The styling department began by crafting small-scale conceptual models of the new prototype. Curved surfaces had to be manually enlarged to full-scale by measuring points offset from these smaller clay mock-ups, then re-plotted onto a life-sized drawing board.
The plotted points were interpolated using French Curves or flexible spline rulers to arrive at a best-fit approximation. Plywood cross sections would be cutout and assembled crosswise, creating a intersecting matrix formwork. The skeleton was fleshed out with clay and further sculpted to create a detailed master model of the car. In its final form, the master model would continue to undergo subtle shaping and refinement at the hands of skilled plasters before being frozen in its final shape, ready for production.
Extracting the relevant geometry and ensuring it fitted seamlessly within the overall assembly would entail similarly onerous techniques of translation. However, the often qualatative interpretation of the geometry introduced significant room for inconsistency, discrepancy, and human error. The company needed a common geometrical language to store each part in terms of numbers, rather than relying on time consuming and error prone process of manual replication.
Fresh from his PhD studies, the young mathematician was tasked with devising a system of equations, so that shapes could be reliably stored and communicated amongst the designers and engineers, as well as input into the primitive computers that drove the new milling machines. De Casteljau struggled with the problem at first.
His epiphany occurred when he realized that, rather than defining a curve by points along its length, it could be accurately plotted and controlled by manipulating a few points around its vicinity. New poles pink connect these points and, once again, points are created at the same proportional distance along them. New poles and points are generated until one final point remains. The process can be repeated at each interval of subdivision along the poles, ad infinitum, to produce a smooth curve.
The formulas describing this differential geometry may be mathematically advanced, but they can be intuitively understood see Figure 6. Points are created at proportional distances ie. New lines connect these points and, once again, a new set of points is created at the same proportional distance along them.
New poles and points are generated until a final point remains. Mathematically, the process can be repeated at all intervals ad infinitum, thanks to computation , plotting a smooth course of curvature.
Upon making his discovery in , de Casteljau immediately rushed to the workshops to set about recreating the the hood of a 2CV from mathematical data alone, only to find that the modelers were unwilling to cooperate. How could a mere ten lines achieve what others before him had unsuccessfully attempted in sixty pages of equations? How could this be a worthy method? If valid, it threatened to replace the black arts of auto production with cold hard computation and, in doing so, make the influential role of master modelers obsolete.
The young mathematician did eventually convince the company that his elegant algorithm was the ultimate solution they had sought by replicating ever more complex auto parts from numerical data, culminating in a full body model of the GS. Welcome or not, the computational curve had arrived.
It was roughly fifteen years before he was personally credited with his discovery. Renault also found the process of enshrining the geometry of each car in a master model and transferring it to the various parts fraught with internal disagreements, retouching, delays, and expense. In addition to curved drawing tools, his department had begun to use simple mathematical constructs, equivalent to cartesian grids onto which regular mathematical curves — arcs, ellipses, and parabolas — were plotted.
Once formulaically enmeshed within the two-dimensional grid, these curves could be morphed by mathematically deforming the grid around them. But when dealing with more complex real-world curves in 3D — the edge of a door panel, for example — it was impossible to apply this method. The fundamental problem remained: Parts of the cylinder outside the cube have omitted for clarity. These geometries could be mathematically defined by encompassing them within a cube left.
By mathematically transforming the cube, the curve inside is also reshaped right. In practice, three vectors pink, blue, yellow between four vertices were only necessary for this construct.
Although this imaginary contraption was impractical, it proved conceptually useful. A 3D parametric curve can be expressed as the intersection of two cylinders see Figure 7. The center and radii points of these cylinders correspond to vertices of a cube, encompassing the curve within.
As he developed the idea, he found only three connected edges of the cube were needed to construct his irregular curves, and that he could formulate them as polynomials. Making more complicated curves, simply required more interconnected lines. The subsequent integration of their geometry into mainstream computing has directly benefitted every design field. Of course a car, being a three dimensional form, is not merely defined by a series of composite curves. Sheet metal body panels, for example, vary in their curvature at almost every point.
Defining only the perimeter curves of each part does not describe its interior contours. Sand is packed into an open-topped wooden box. Two of the opposing top edges of the box are curved so that when a wooden plank or template is dragged across them, it scrapes off the remaining sand to leave a smoothly curved surface.
Taking a grid of points black , he established a set of curves in one direction yellow. Points blue at equal proportions along these rails form curves traveling in the opposite direction pink. Recursively sweeping across the rails at every proportional distance defines a surface. Again, he formulated a mathematical construction of his hypothetical tooling see Figure 9.
Between these, another pair of curves is inserted to yield four unique curves in one direction.
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SLS Undergraduates Michelle Akamine and Victoria Lee Receive Research Awards for Projects on Child Second Language Acquisition Drs. Kyle, LaFlair, and Ziegler Awarded TOEFL COE Grant PhD Student Awarded NSF Doctoral Dissertation Research Grant Develop Your Expertise as a Language Teacher Learn Cutting Edge Research Methods Learn a New Language While You Earn Your Degree SLS . The University of Iowa shooting took place in Iowa City, Iowa, United States on November 1, The gunman was Gang Lu, a year-old former graduate student at the University. He killed four members of the university's faculty and a student; he left another student seriously injured, before committing suicide.