We begin by expressing 𝑧 in terms of 𝑤 as follows: We can represent this visually as follows.Įxample 5: Composite Transformations of the Complex Planeįind an equation for the image of | 𝑧 − 2 | = 3 under the transformation of 𝑧-plane The origin by 𝜋 2 radians, and 𝑇 , a translation by the vector We can interpret this transformation as the combination of two transformations 𝑇 , a counterclockwise rotation about This represents a half-line whose end point is at 3 + 3 𝑖 which makes an angle of 𝜋 with the positive horizontal. Using the properties of the argument, we can rewrite this asĪ r g a r g ( 𝑤 − 2 − 3 𝑖 ) − ( 𝑖 ) = 𝜋 2. Įxpressing the subject of the argument as a single fraction, we haveĪ r g 𝑤 + 3 + 𝑖 − 4 𝑖 + 5 𝑖 𝑖 = 𝜋 2
We can find an equation for the image of a r g ( 𝑧 − 4 + 5 𝑖 ) = 𝜋 2 by first expressing 𝑧 in terms of 𝑤 We are considering a transformation 𝑇 from the 𝑧-plane to another complex plane which we will call Which is a counterclockwise rotation about the origin through an angle of 𝜃.Įxample 4: Compound Transformations of the Complex Planeįind an equation for the image of the half-lineĪ r g ( 𝑧 − 4 + 5 𝑖 ) = 𝜋 2 under the transformation 𝑇 ∶ 𝑧 ↦ 𝑖 𝑧 − 3 − 𝑖. 𝑟, and the transformation we get by multiplying by 𝑒 , Get by multiplying by the real number 𝑟, which is a dilation by scale factor To gain a clearer understanding of what a given transformation is doing.Ĭombining the results of examples 2 and 3, we can see that multiplication by a generalĬomplex number 𝑧 = 𝑟 𝑒 can be understood as a combination of the transformation we This is why we often need to consider the image of both lines and circles This would not have been very insightful. We would have found that its image would be a circle in the 𝑤-plane with the same center and radius. Notice that had we considered the image of a circle in the previous example, Moreover, we know that it did not scale the line since the modulus of 𝑒 is one. This demonstrates that the transformation rotated the plane by 𝑤 = 𝑒 𝑧 , the image half-line at the origin which makes anĪngle of 𝜃 to the real axis is a half-line at the origin which makes anĪngle of 𝜃 + 𝜑. This gives us the the y axis along the tangents, the x axis along the binormals and the z axis along the normals.The previous example showed us that, under the transformation Placing the x axis of the cone along the normals of the path would leave the z axis pointing in the opposite direction to the path binormals. Our example requires the y axis of the cone to lie along the tangents of the path. To help undertand which of the 6 possible arrangements it is useful to show the axes of the mesh that you want to follow the line and the tangents, normals and binormals of the path.įor example, we want a cone to follow along a path point first and with the height axes of the cone tangental to the path.īelow is an image showing the cone axes and the path tangents, normals and binormals at the start of the path before any alignment with the path takes place Aligning a Plane To a Curve Refining Alignment Others can twist the plane at certain points. The top one has the plane tangential to the curve and the fourth one down is perpendicular to the curve. There are 6 ways to order a group of three axes and so 6 ways to align the plane axes to the curve point axes.Īll six ways are used in the playground below.
Using the RotationFromAxis we can align the x, y and z axes of the plane with the point axes of the curve. By creating a 3D path from this curve we can obtain the normal, tangent and binormal of the curve at each of the positions that define it.
We can draw a curve in space using an array of position vectors. A plane created in Babylon.js has a normal along the z axis with the x and y axes lying in the plane. Where axis1, axis2 and axis3 are three left-handed orthogonal vectors and the mesh will be aligned withĪt any point along curve in 3D space the tangent, normal and binormal form a set of orthogonal axes, call these the point axes.
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