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A local transform specifies the local rotation and position of a Spatial inside its parent.So:transform.basis.x|y|z will return the axis-vectors rotated by the local rotation.transform.origin will return the local offset of a Spatial inside its parent.

This methods calculates the transform.basis (rotation) matrix which is needed to let the forward vector (Vector3(0,0,-1)) point from Transform.origin to the target coordinates. The "up" Vector defines the orientation of the resulting Transform.basis.

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Example:You have a node with a "head" mesh. When you let that head "look_at" another node and you pass a down-vector (Vector3(0,-1,0)) then the hair will be below the head. If you pass an up-vector (Vector3(0,1,0)) then the hair will be on the upside of the head. In any way the head will always point in the right direction, only the orientation is different.

Say you have vectors p0 and p1. The midpoint between them is simply (p0+p1)/2. More generally, the line segment defined by p0 and p1 can be generated by varying t between 0 and 1 in the following linear interpolation:

Because the dot product between a and b is also defined as |a||b|cos(alpha) (where alpha is the angle between the two), the result will be 0 if the two vectors are perpendicular, positive if the angle between them is less than 90, and negative if greater. This can be used to tell if two vectors point in the same general direction.

Say that you have a list of directions represented as unit vectors, and you want to find which of them is the closest to dir. Simply find the largest dot product between dir and a vector in the list. Likewise, the smallest dot product will be the direction farthest away.

Say that you have an arbitrary point in space, p0, and a direction (unit) vector, dir. Imagine that an infinite line goes by p0, perpendicular to dir, dividing the plane in two, the half-plane that dir points to, and the half-plane that it does not point to. How do I tell whether a point p is in the side pointed to by dir? Remember that dot product is positive when the angle between vectors is less than 90 degrees, so just project and check against that:

I'm an engineering-physics student taking a course in complex analysis, and it's a little frustrating, because I see all these connections to vector calculus over the reals (especially as applied to electromagnetics). For example, if we treat complex numbers like vectors, then the Cauchy-Riemann equations tell us that the derivative exists if the divergence and curl are both zero, which is equivalent to saying that our field is conservative, and there's no field sources/sinks. That these functions also satisfy Laplace's equation is also fairly obvious, since Laplace's equation describes a conservative field with no field sources.

Essentially, to what extent can we pretend the complex numbers are vectors, and bring over useful results from vector calculus? If this doesn't work, then what are the major flaws, or things to watch out for?

Edit: The reason I'm looking for analogs between vector analysis and complex analysis is because results in vector analysis usually have clear physical interpretations. I'm hoping to enhance my understanding of complex analysis through the similarities. Additionally, if I know a proof for certain proprerties of vector functions, I probably shouldn't focus my attention on the proof of an equivalent property in complex analysis. Instead, I'd prefer to focus attention on results that don't have a correspondence to vector analysis.

So I'm wondering, what are some results in complex analysis that don't have a similar result in vector analysis? Or even for results that are similar, what are the special peculiarities to complex analysis that make the results different?

I think we do not really cheat in complex analysis. If I recall correctly, one of the popular proofs of Cauchy's integral formula is to use Green's theorem assuming the region is good enough. I think the reason you observe these things is because complex functions are harmonic. So tools in PDE and functional analysis naturally enters into the stage. Similarly you can prove an analytical function is infinitely many times differentiable by using maximal principle. But I think in real life the situation is the other way around, that we use complex functions as representations of "nicest" functions possible. And there is a geometric side which is not captured by the analytical nature of harmonic functions. For example a lot of complex functions are conformal, and understanding how it acts on the plane is a big deal to number theorists and analysts alike.

This article is focused on providing application security testing professionals with a guide to assist in Cross Site Scripting testing. The initial contents of this article were donated to OWASP by RSnake, from his seminal XSS Cheat Sheet, which was at: That site now redirects to its new home here, where we plan to maintain and enhance it. The very first OWASP Prevention Cheat Sheet, the Cross Site Scripting Prevention Cheat Sheet, was inspired by RSnake's XSS Cheat Sheet, so we can thank RSnake for our inspiration. We wanted to create short, simple guidelines that developers could follow to prevent XSS, rather than simply telling developers to build apps that could protect against all the fancy tricks specified in rather complex attack cheat sheet, and so the OWASP Cheat Sheet Series was born.

This cheat sheet lists a series of XSS attacks that can be used to bypass certain XSS defensive filters. Please note that input filtering is an incomplete defense for XSS which these tests can be used to illustrate.

Originally found by Begeek (but cleaned up and shortened to work in all browsers), this XSS vector uses the relaxed rendering engine to create our XSS vector within an IMG tag that should be encapsulated within quotes. I assume this was originally meant to correct sloppy coding. This would make it significantly more difficult to correctly parse apart an HTML tags:

Some websites claim that any of the chars 09-13 (decimal) will work for this attack. That is incorrect. Only 09 (horizontal tab), 10 (newline) and 13 (carriage return) work. See the ascii chart for more details. The following four XSS examples illustrate this vector:

Null chars also work as XSS vectors but not like above, you need to inject them directly using something like Burp Proxy or use %00 in the URL string or if you want to write your own injection tool you can either use vim (^V^@ will produce a null) or the following program to generate it into a text file. Okay, I lied again, older versions of Opera (circa 7.11 on Windows) were vulnerable to one additional char 173 (the soft hyphen control char). But the null char %00 is much more useful and helped me bypass certain real world filters with a variation on this example:

Submitted by Franz Sedlmaier, this XSS vector could defeat certain detection engines that work by first using matching pairs of open and close angle brackets and then by doing a comparison of the tag inside, instead of a more efficient algorithm like Boyer-Moore that looks for entire string matches of the open angle bracket and associated tag (post de-obfuscation, of course). The double slash comments out the ending extraneous bracket to suppress a JavaScript error:

In Firefox and Netscape 8.1 in the Gecko rendering engine mode you don't actually need the \> portion of this Cross Site Scripting vector. Firefox assumes it's safe to close the HTML tag and add closing tags for you. How thoughtful! Unlike the next one, which doesn't effect Firefox, this does not require any additional HTML below it. You can add quotes if you need to, but they're not needed generally, although beware, I have no idea what the HTML will end up looking like once this is injected:

Unlike Firefox the IE rendering engine doesn't add extra data to you page, but it does allow the javascript: directive in images. This is useful as a vector because it doesn't require a close angle bracket. This assumes there is any HTML tag below where you are injecting this cross site scripting vector. Even though there is no close ">" tag the tags below it will close it. A note: this does mess up the HTML, depending on what HTML is beneath it. It gets around the following NIDS regex: /((\\%3D)|(=))\[^\\n\]\*((\\%3C)|\)/ because it doesn't require the end ">". As a side note, this was also affective against a real world XSS filter I came across using an open ended

When the application is written to output some user information inside of a JavaScript like the following: var a="$ENV{QUERY\_STRING}"; and you want to inject your own JavaScript into it but the server side application escapes certain quotes you can circumvent that by escaping their escape character. When this gets injected it will read var a="\\\\";alert('XSS');//"; which ends up un-escaping the double quote and causing the Cross Site Scripting vector to fire. The XSS locator uses this method.:

Fairly esoteric issue dealing with embedding images for bulleted lists. This will only work in the IE rendering engine because of the JavaScript directive. Not a particularly useful cross site scripting vector:

Using something as simple as a remote style sheet you can include your XSS as the style parameter can be redefined using an embedded expression. This only works in IE and Netscape 8.1+ in IE rendering engine mode. Notice that there is nothing on the page to show that there is included JavaScript. Note: With all of these remote style sheet examples they use the body tag, so it won't work unless there is some content on the page other than the vector itself, so you'll need to add a single letter to the page to make it work if it's an otherwise blank page:

This works the same as above, but uses a tag instead of a tag). A slight variation on this vector was usedto hack Google Desktop. As a side note, you can remove the end tag if there is HTML immediately after the vector to close it. This is useful if you cannot have either an equals sign or a slash in your cross site scripting attack, which has come up at least once in the real world: ff782bc1db