![]() Since vectors represent directions, the origin of the vector does not change its value. Because it is more intuitive to display vectors in 2D (rather than 3D) you can think of the 2D vectors as 3D vectors with a z coordinate of 0. If a vector has 2 dimensions it represents a direction on a plane (think of 2D graphs) and when it has 3 dimensions it can represent any direction in a 3D world.īelow you'll see 3 vectors where each vector is represented with (x,y) as arrows in a 2D graph. Vectors can have any dimension, but we usually work with dimensions of 2 to 4. The directions for the treasure map thus contains 3 vectors. You can think of vectors like directions on a treasure map: 'go left 10 steps, now go north 3 steps and go right 5 steps' here 'left' is the direction and '10 steps' is the magnitude of the vector. You should assume this, unless it is noted in the problem that you need to rotate clockwise. Conventionally, shapes are rotated counterclockwise on a coordinate plane. Rotating a shape 270 degrees is the same as rotating it 90 degrees clockwise. A vector has a direction and a magnitude (also known as its strength or length). Note the corresponding clockwise and counterclockwise rotations. ![]() In its most basic definition, vectors are directions and nothing more. If the subjects are difficult, try to understand them as much as you can and come back to this chapter later to review the concepts whenever you need them. The focus of this chapter is to give you a basic mathematical background in topics we will require later on. However, to fully understand transformations we first have to delve a bit deeper into vectors before discussing matrices. When discussing matrices, we'll have to make a small dive into some mathematics and for the more mathematically inclined readers I'll post additional resources for further reading. Matrices are very powerful mathematical constructs that seem scary at first, but once you'll grow accustomed to them they'll prove extremely useful. This doesn't mean we're going to talk about Kung Fu and a large digital artificial world. There are much better ways to transform an object and that's by using (multiple) matrix objects. We could try and make them move by changing their vertices and re-configuring their buffers each frame, but that's cumbersome and costs quite some processing power. We now know how to create objects, color them and/or give them a detailed appearance using textures, but they're still not that interesting since they're all static objects. When plot these points on the graph paper, we will get the figure of the image (rotated figure).Transformations Getting-started/Transformations In the above problem, vertices of the image areħ. When we apply the formula, we will get the following vertices of the image (rotated figure).Ħ. When we rotate the given figure about 90° clock wise, we have to apply the formulaĥ. ![]() When we plot these points on a graph paper, we will get the figure of the pre-image (original figure).Ĥ. In the above problem, the vertices of the pre-image areģ. First we have to plot the vertices of the pre-image.Ģ. So the rule that we have to apply here is (x, y) -> (y, -x).īased on the rule given in step 1, we have to find the vertices of the reflected triangle A'B'C'.Ī'(1, 2), B(4, -2) and C'(2, -4) How to sketch the rotated figure?ġ. Here triangle is rotated about 90 ° clock wise. If this triangle is rotated about 90 ° clockwise, what will be the new vertices A', B' and C'?įirst we have to know the correct rule that we have to apply in this problem. Let A(-2, 1), B (2, 4) and C (4, 2) be the three vertices of a triangle. Let us consider the following example to have better understanding of reflection. Here the rule we have applied is (x, y) -> (y, -x). ![]() How to Describe a Rotation Describing a rotation is easy. To find B, extend the line AB through A to B’ so that. In this case, since A is the point of rotation, the mapped point A’ is equal to A. A point that rotates 180 degrees counterclockwise will map to the same point if it rotates 180 degrees clockwise. The image below shows a shape rotated by an angle of 120° clockwise. Because the given angle is 180 degrees, the direction is not specified. The direction of the rotation (clockwise or anti-clockwise) can also be described. It can be described in degrees or radians. Once students understand the rules which they have to apply for rotation transformation, they can easily make rotation transformation of a figure.įor example, if we are going to make rotation transformation of the point (5, 3) about 90 ° (clock wise rotation), after transformation, the point would be (3, -5). The Angle of Rotation The angle of rotation is the angle that the shape has been rotated about.
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