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Areas are usually bounded by closed polylines or polygons. A sequence of line where the following line starts where the previous one ends is called a polyline.
For various applications—for instance for modelling surfaces—additional properties of polygons are required.
One of such properties is that the polygon should not overlap with itself. Convexity is another important property that is often needed. A polygon or, more generally, an area or a region is convex if whenever two points are within the region the 2. For the nonconvex polygon two points inside the polygon are chosen and connected by a dotted line that lies not completely inside the polygon. In addition to lines and piecewise linear polylines, curves are also common in computer graphics.
In general, it will not pass through control points.
In the case of a quadratic curve with one control point one can imagine the lines connecting the control point with the two endpoints. The connecting lines are the tangents of the quadratic curve in the two endpoints. The quadratic curve is given by two endpoints and one control point through which the curve does not pass.
The tangents in the endpoints are also shown here as dotted lines. The resulting joint curve would be continuous, but not smooth, i.
In order to avoid sharp bends, the tangent of the endpoint of the previous curve and the 16 2. Basic principles of two-dimensional graphics following curve must point into the same direction. This means they must lie on the same line. Areas are bounded by a closed curve.
Axes-parallel rectangles, whose sides are parallel to the coordinate axes, play an important role in computer graphics. The union joins two areas to a larger area whereas their intersection consists of the part belonging to both areas.
They provide another way of constructing new areas from already existing ones. The abstract class Shape with its various subclasses allows the construction of various twodimensional geometric objects.
Shapes will not be drawn until the draw or the fill method is called with the corresponding Shape as argument in the form graphics2d. The abstract class Point2D for points is not a subclass of Shape.
Points cannot be drawn directly. If one wants to draw a point, i. Objects of the class Point2D are mainly used to specify coordinates for other geometric objects.
Therefore, the class Point2D will not occur very often in the example programs. The abstract class Point2D is extended by the two classes Point2D. Float and Point2D. When using the abstract class Point2D it is not necessary to specify whether coordinates are given as float- or double-values. The same concept is also used for most of the other geometric objects. Basic principles of two-dimensional graphics The elementary geometric objects in Java 2D introduced in the following extend the class Shape, so that they can be drawn by applying one of the methods draw or fill.
Double x1,y1,x2,y2 ; The parameters x1, y1, x2 and y2 are of type double. Similarly, Line2D.
Float requires the same parameters, but of type float. Only when the method g2d. Analogously to lines, quadratic curves are modelled by the abstract class QuadCurve2D. Java 2D provides the abstract class CubicCurve2D for modelling cubic curves.
Analogously to the cases of lines and quadratic curves, CubicCurve2D. Double x1,y1, ctrlx1,ctrly1, ctrlx2,ctrly2, x2,y2 ; The program CurveDemo. Double, QuadCurve2D.
Double and CubicCurve2D. The class GeneralPath allows the construction not only of polylines, i. A GeneralPath starts in the origin of the coordinate system, i. Each method will append a corresponding line or curve to the endpoint of the last element in the sequence of the GeneralPath.
The methods lineTo, quadTo and curveTo append a line, 2. In addition to these three methods for curves and lines, the class GeneralPath also contains the method moveTo that allows to jump from the endpoint of the previous curve to another point without connecting the points by a line or curve. Basic principles of two-dimensional graphics g2d. The complete class for drawing the car can be found in the example program GeneralPathCar.
In addition to the class GeneralPath Java 2D also provides classes for axes-parallel rectangles and ellipses as basic geometric objects. By the class Rectangle2D. It is still necessary to call the method g2d. A circle is a special case of an ellipse, where the bounding rectangle is a square. A circle with centre point x, y and radius r can be generated by Ellipse2D.
The angle is given as a float-value in terms of degrees. Otherwise, the angle is determined relative to the rectangle. Analogously to the start angle, extend corresponds to the true angle of the arc only in the case of a circular arc.
PIE and Arc2D. CHORD, specifying whether only the arc itself, the corresponding segment or the arc with the chord of the ellipse, respectively, should be constructed.
Basic principles of two-dimensional graphics Figure 2. The same applies to the opening angle. At the end of section 2. Given two Area objects areaA and areaB, the following methods are available, implementing the corresponding set-theoretic operations. The Area object areaA contains the result of the application of the corresponding set-theoretic operation. Geometric transformations can be used to position objects, i. Before discussing geometric transformations in more detail, it is necessary to explain some general conventions.
In computer graphics, points as well as vectors are used. From a purely mathematical point of view, both can be represented as elements of the space Rn , i. Especially in physics, it is very important to distinguish clearly between these two concepts of points and vectors. A tuple x1 ,. Hopefully, physicists will tolerate the abuse of notation in the context of this book. For equations within this book, column vectors will be used consistently. Within the text, points are sometimes written as row vectors in order to avoid stretching of text lines.
In those cases where a point is explicitly understood as a column vector, the symbol for transposing vectors will be used, i. The dot product of two vectors u and v will be denoted in the following 24 2. A scaling leads to stretching or shrinking of objects in the direction of the x- and the y-axis.
The same holds for all other geometric transformations. They carry out pointwise transformations of objects. The 2. But in addition, the centre of the rectangle is also transformed so that the transformed rectangle is shifted to the lower right compared to the original rectangle. A scaling is always carried out with respect to the origin of the coordinate system. Students will learn the fundamentals of Java. The focus is on developing high quality, working software that solves real problems.
Prerequisites: Designed for students with some programming experience. The idea is that by thinking about mathematical problems, students are prodded into learning MATLAB for the purpose of solving the problem at hand. Topics include variables, arrays, conditional statements, loops, functions, and plots. Prerequisites: There are no formal prerequisites for this course.
Topics include variables, scripts, and operations; visualization, solving equations, and curve fitting; and Simulink. Prerequisites: Basic familiarity with programming; basic linear algebra, differential equations, and probability.
You will learn the required background knowledge, including memory management, pointers, preprocessor macros, object-oriented programming, and how to find bugs when you inevitably use any of those incorrectly. Students learn how to write software that is safe from bugs, easy to understand, and ready for change.
It covers the common algorithms, algorithmic paradigms, and data structures used to solve these problems. The course emphasizes the relationship between algorithms and programming, and introduces basic performance measures and analysis techniques for these problems. Prerequisites: A firm grasp of Python and a solid background in discrete mathematics are necessary prerequisites to this course. You are expected to have mastered the material presented in 6.
Using Java, student teams program virtual robots to play Battlecode, a real-time strategy game. Optional lectures are provided on topics and programming practices relevant to the game, and students learn and improve their programming skills experientially. The course culminates in a live tournament. Prerequisites: Experience in programming definitely helps in the competition.