Reflection over y = xĪ reflection across the line y = x switches the x and y-coordinates of all the points in a figure such that (x, y) becomes (y, x). All of the points on triangle ABC undergo the same change to form DEF. Triangle DEF is formed by reflecting ABC across the y-axis and has vertices D (4, -6), E (6, -2) and F (2, -4). ![]() Algebraically, the ordered pair (x, y) becomes (-x, y). In a reflection about the y-axis, the y-coordinates stay the same while the x-coordinates take on their opposite sign. Triangle DEF is formed by reflecting ABC across the x-axis and has vertices D (-6, -2), E (-4, -6) and F (-2, -4). Algebraically, the ordered pair (x, y) becomes (x, -y). In a reflection about the x-axis, the x-coordinates stay the same while the y-coordinates take on their opposite signs. The most common cases use the x-axis, y-axis, and the line y = x as the line of reflection. There are a number of different types of reflections in the coordinate plane. This is true for any corresponding points on the two triangles and this same concept applies to all 2D shapes. A, B, and C are the same distance from the line of reflection as their corresponding points, D, E, and F. The figure below shows the reflection of triangle ABC across the line of reflection (vertical line shown in blue) to form triangle DEF. The same is true for a 3D object across a plane of refection. In a reflection of a 2D object, each point on the preimage moves the same distance across the line of reflection to form a mirror image of itself. The term "preimage" is used to describe a geometric figure before it has been transformed "image" is used to describe it after it has been transformed. When an object is reflected across a line (or plane) of reflection, the size and shape of the object does not change, only its configuration the objects are therefore congruent before and after the transformation. In geometry, a reflection is a rigid transformation in which an object is mirrored across a line or plane. To scale the shape by a factor of 3, we scale its coordinates by a factor of 1/3.Home / geometry / transformation / reflection ReflectionĪ reflection is a type of geometric transformation in which a shape is flipped over a line. The scaled-up circle has the equation sqrt((x/3)^2 + (y/3)^2) - 1mm = 0mm. 3 We want to scale our unit circle by a factor of 3. Input a Subtract block into the X inputĮx.The blue circle is the translated object.Įxample as shown using a primitive sphere block. To move the circle +1 in the x-direction, we replaced x by x-1, not by x+1. We want to shift this circle by +1 unit along the x-axis, the new equation is sqrt((x-1)^2 + y^2) - 1mm = 0mm. 2 The unit circle centered at the origin has the equation sqrt(x^2 + y^2) - r = 0. 0 = z - f(x, y)You want the expression opposite the zero to be negative where the part is solid.Įx. If you want to plot a function in the form z = f(x, y), you can implicitize it by moving all the terms to one side. Understanding Remapping through Equations If your object is at the origin, and you move the origin (-1, 0, 0), your object ends up at (1, 0, 0) after. To apply a transformation to a shape, you have to apply the inverse transformation to its coordinate system. One main difference between explicit and implicit modeling is that explicit geometry transforms actively (you move it where you want it), while implicits transform passively (you move the coordinate system, not the object). When you add and multiply field values in nTop, you're not directly modifying a shape, you're modifying a coordinate system used to represent that shape. Multiplication scales the model while addition and subtraction translate the model. We stretch all the X values but keep the same Y and Z values of the sphere. X: X*10 (can also use a Multiply block).1 Let's use the remap block to magnify a sphere. This is what happens with Remap Field, but we do it to 3D geometry across XYZ.Įx. Basically, you are stretching out these values. If you multiply this number by 10, you get. ![]() In general, Remap Field allows you to warp geometry by supplying functions or fields to specify a replacement position for every point in the model. To gain a further understanding of fields and remapping in nTop, take a look at this Field-Driven Design White Paper by George Allen, an nTop Fellow. You can translate or scale an object with other blocks in nTop, but this method will help you understand how remapping works. Learn how to remap a field to scale or translate an implicit body by using blocks and equations.
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