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The term spline refers to an equation or series of equations that describes a curve or set of objects. For example, the equation for the circle is y=x2, which can be represented as 3 * x + 4 for its three parameters and then 5 for the value of y. This tutorial will show how to use t splines in Rhino 5.0 by teaching you how to derive a t spline using simple arithmetic operations and then finding its tangents at one point. What are t-splines? A T - shaped object can be represented with two line segments, one where it attaches at one end and another where it attaches at the other end. The 3D T spline can be drawn using two line segments, one for the top where it joins the 2D version and one for the bottom. There are many ways to draw splines in Rhino. This tutorial will use the simplest way which is to use T - shaped objects. This tutorial uses splines drawn with lines drawn by hand, but they can also be drawn automatically by Rhino commands. What are splines good for? Splines are typically drawn with control points that help control or create a curve or surface. A typical example of this is drawing a curve around a portion of an object to cut out holes in objects. Splines are typically not used for orthogonal geometry, but can be used to generate mitered joints or complex curves. A Miter joint is a joint made between two planar objects with an angle so that the ends of one object aligns with its adjacent object.The t-spline has 3 control points, one at each end of the spline. For this example, we will use control points that are positioned equally along the spline. The t-spline equation is derived using simple arithmetic operations on the control points' coordinates. The first step is to combine like terms on each end of the spline to simplify the equations (see "Combining like terms"). The coordinates of the first control point is given by x1, y1 and then x2 and y2 with z2 as a constant: The second and third equations are similar. Neither of these equations actually represents the t-spline correctly. It would be more appropriate to derive an equation which looks like this: To obtain this equation we need to combine like terms on each end of the spline as follows: (x - x0) * (y - y0) + (x0 * x + y0 * y) + ((x0 - 1)(y - 1)) = 0 Where 0 < x, y < 1 and 0 < x, y < 1 . The equations are simplified by inserting the value of z2 as a constant. For example, for equation 1, we could replace x1 + y1 with x1 + z2. 81eaaddfaf 44
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