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types:vector [2015-01-13 18:39 SLT] sei Subtle style change |
types:vector [2015-09-29 08:15 SLT] (current) sei Fix cross product expression |
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| ===== Vector type ===== | ===== Vector type ===== | ||
| - | Unlike many other languages, LSL has native support for several operations that are common in 3D geometry. A type that is fundamental in that support is the **vector** type. At the lowest level, a vector can be defined as a type that is able to hold three floats; but LSL also supports many operations that are designed for geometry handling. | + | Unlike many other languages, LSL has native support for several operations that are common in 3D geometry. A type that is fundamental in that support is the $ty[vector] type. A vector can be defined as a type that is able to hold three floats; but LSL also supports many operations that are designed for geometry handling. |
| - | A vector literal is written in LSL as three comma-separated [[float]] values within angle brackets (e.g. ''<1.5, 0.0, -7.3>'' is a valid vector). As in many other instances, if an integer appears instead of a float, it is automatically cast to a float, therefore this is also a valid vector literal: ''<0, 3, -5>'' which represents the vector ''<0.0, 3.0, -5.0>''. | + | A vector literal is written in LSL as three comma-separated $lty[float] values within angle brackets (e.g. ''<1.5, 0.0, -7.3>'' is a valid vector). As in many other instances, if an integer appears instead of a float, it is automatically cast to a float, therefore this is also a valid vector literal: ''<0, 3, -5>'' which represents the same vector as ''<0.0, 3.0, -5.0>''. |
| - | The components of a vector can be referenced only when they are inside variables, and the syntax is the name of the variable followed by a dot and then the letters **x**, **y** or **z**. For example, if ''vector_var'' is a variable of type **vector**, then ''vector_var.y'' is the Y component of the vector. | + | The components of a vector can be referenced only when they are inside variables, and the syntax is the name of the variable followed by a dot and then the letters **x**, **y** or **z**. For example, if ''vector_var'' is a variable of type **vector**, then ''vector_var.y'' is the second (**y**) component of the vector. |
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| + | Due to their ability to store three floats, vectors are sometimes "abused" to hold three values even if they don't make a vector in the geometric sense. For example, they are used to store the Euler angles of a rotation, or the RGB components of a colour. Sometimes even one of the values is discarded, such as in the case of a Path Cut specification, which is written in a vector whose x component is the path cut begin, the y component is the path cut end, and the z component is ignored. | ||
| === Negation of vectors === | === Negation of vectors === | ||
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| === Addition and subtraction of vectors === | === Addition and subtraction of vectors === | ||
| - | Vectors can be added, resulting in a new vector whose components are the sum of the corresponding component of each vector. The symbol is the same as for addition of floats and integers, i.e. the plus sign **+**. For example, if **v** and **w** are vector variables, then ''v + w'' gives the same result as ''<v.x + w.x, v.y + w.y, v.z + w.z>''. Subtraction works similarly using the **-** symbol instead. | + | Vectors can be added, resulting in a new vector whose components are the sum of the corresponding component of each vector. The symbol is the same as for addition of floats and integers, i.e. the plus sign $op[+]. For example, if **v** and **w** are vector variables, then ''v + w'' gives the same result as ''<v.x + w.x, v.y + w.y, v.z + w.z>''. |
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| + | Subtraction works similarly using the $op[-] symbol instead. | ||
| === Multiplication of vectors === | === Multiplication of vectors === | ||
| - | Vectors can be multiplied in several ways. The most basic one is multiplication by a float: that returns a vector where each of its components is that of the original vector multiplied by the given float. As usual, integers can be used in place of floats and will be automatically cast to float. For example, ''<1, 2, 3> * 5'' gives the same vector as ''<5, 10, 15>''. | + | Vectors can be multiplied in several ways. The most basic one is multiplication by a float: that returns a vector where each of its components is that of the original vector multiplied by the given float. As usual, integers can be used in place of floats and will be automatically cast to float. For example, ''<1, 2, 3> * 5'' and ''5 * <1, 2, 3>'' both give the same vector as ''<5, 10, 15>''. |
| - | Another type of multiplication is the so called [[http://en.wikipedia.org/wiki/Dot_product|dot product or scalar product]]. It takes two vectors, and the result is a //float//. The symbol is the same as for multiplication of floats and integers, i.e. the asterisk **`***. The resulting operation is the sum of the products of the components; for example, if **v** and **w** are vector variables, then ''v*w'' gives the same result as ''v.x*w.x + v.y*w.y + v.z*w.z''. It is useful in several geometric applications. | + | Another type of multiplication is the so called [[http://en.wikipedia.org/wiki/Dot_product|dot product or scalar product]]. It takes two vectors, and the result is a //float//. The symbol is the same as for multiplication of floats and integers, i.e. the asterisk $op[*]. The resulting operation is the sum of the products of the components; for example, if **v** and **w** are vector variables, then ''v * w'' gives the same result as ''v.x * w.x + v.y * w.y + v.z * w.z''. It is useful in several geometric applications. |
| - | A third type of multiplication is called [[http://en.wikipedia.org/wiki/Cross_product|cross product or vector product]]. It takes two vectors, and the result is a //vector//. The symbol is the same as for integer modulo, i.e. the percent sign **%**. The operation is a bit complex: if **v** and **w** are vector variables, then ''v%w'' gives the same result as ''<v.y*w.z - v.z*w.y, v.z*w.x - v.x*w.z, v.x*w.z - v.z*w.x>''. This kind of multiplication returns a vector that is perpendicular to both input vectors, and has several uses in geometry applications too. | + | A third type of multiplication is called [[http://en.wikipedia.org/wiki/Cross_product|cross product or vector product]]. It takes two vectors, and the result is a //vector//. The symbol is the same as for integer modulo, i.e. the percent sign $op[%]. The operation is a bit complex: if **v** and **w** are vector variables, then ''v % w'' gives the same result as ''<v.y*w.z - v.z*w.y, v.z*w.x - v.x*w.z, v.x*w.y - v.y*w.x>''. This kind of multiplication returns a vector that is perpendicular to both input vectors, and has several uses in geometry applications too. |
| - | And finally, there is a fourth type of multiplication: a vector multiplied by a [[rotation]] has the effect of applying the rotation to that vector, returning the rotated vector. For example, if **r** is a rotation that turns 90° counter-clockwise over the positive Z axis, then ''<0, 3, 1> * r'' will result in ''<-3, 0, 1>'', that is, the vector will be rotated over that axis by that angle. | + | And finally, there is a fourth type of multiplication: a vector multiplied by a $lty[rotation] returns the vector rotated by the given rotation. For example, if **r** is a rotation that turns 90° counter-clockwise over the positive Z axis, then ''<0, 3, 1> * r'' will result in ''<-3, 0, 1>'', that is, the vector will be rotated over that axis by that angle. |
| - | Note that there is no operation that returns the product of the components one by one similarly to vector addition. This operation has no meaning in geometry, and is thus not implemented; however, if you need it, it's easy to write like this: say **v** and **w** are your vectors; then you can obtain that result using ''<v.x*w.x, v.y*w.y, v.z*w.z>''. | + | <wrap important>Important Note:</wrap> There is no operation that returns the product of the components one by one similarly to vector addition. This operation has no meaning in geometry, and is thus not implemented; however, if you need it, it's easy to write like this: say **v** and **w** are your vectors; then you can obtain that result using ''<v.x*w.x, v.y*w.y, v.z*w.z>''. |
| === Division of vectors === | === Division of vectors === | ||
| - | Nothing can be divided by a vector, but vectors can be divided by two kind of elements: floats, and rotations. As with multiplication by a float, dividing a vector by a float results in another vector where each element is divided by that float, e.g. ''v/f'' gives the same result as ''<v.x/f, v.y/f, v.z/f>''. Division of a vector by a rotation results in applying to it the inverse rotation to the given one (e.g. if the rotation turns counter-clockwise, the vector will be rotated clockwise if it's divided by it). | + | Nothing can be divided by a vector, but vectors can be divided by two types of elements: floats, and rotations. As with multiplication by a float, dividing a vector by a float results in another vector where each element is divided by that float, e.g. ''v/f'' gives the same result as ''<v.x/f, v.y/f, v.z/f>''. Division of a vector by a rotation results in applying the rotation that cancels the given one (e.g. if the rotation turns counter-clockwise over a certain axis, the vector will instead be rotated clockwise over that axis if it's divided by it). |
| === Type casting of vectors === | === Type casting of vectors === | ||
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| * ''(vector)vector_value'' does nothing special to the value, as expected. | * ''(vector)vector_value'' does nothing special to the value, as expected. | ||
| - | * ''(vector)string_value'' is supported; see [[string#Type casting of strings]] for details. | + | * ''(vector)string_value'' is supported; see $lty[string#Type casting of strings] for details. |
| - | * ''(string)vector_value'' is supported too; see [[float#Precision]] for details. | + | * ''(string)vector_value'' is supported too; see $lty[float#Type casting a float to string] for details. |
| - | * ''(list)vector_value'' results in a list with one single element of type **vector**. | + | * ''(list)vector_value'' results in a list with one single element of type $ty[vector]. |