Posts from the “Pixar in a Box” Category

Proof of the Touching Point Equation of Parabolas

parabolas

Screenshot of Pixar in a Box (Bonus: completing the proof, Environment Modeling)

The proof of the touching point equation in the environment modeling tutorial of Pixar in a Box is very neat, but it took me a second round to understand. Here I’m writing a note hoping making the proof intuitively understandable.

We need to prove that the touching point P of the parabolas arc and line QR is controlled by t on line QR.

The proof starts with writing P as controlled by a random value s on line QR, where s controls line Q’R’. Then, reforming the equation of P controlled by s on line QR results in another equation, in which P is controlled by t on line Q’R’.

The result of the reformation indicates that P lies on line Q’R’ too. Hence, P is the intersection point of QR and Q’R’.

From the observation of section 4, P becomes the touching point of QR when s is equal to t.

There’s no proof of the observation though. But it’s a very neat one that I want to explain. Now that for any point P in QR, P is either the intersection point with line Q’R’, or the touching point. If s doesn’t equal to t, we can always define the intersection point. Otherwise, P has to be the touching point, because we cannot define this one single intersection point.

Proof completed! DAH!

 

Color Science

Spectrum of Light

> What would the spectrum of laser look like?

The spectrum of laser varies based on the type of the laser. Lasers differ to each other by the operation wavelengths. Since most lasers [1] have one or a few operation wavelengths, the spectrum of laser should strike at the operation wavelengths.

> Why is the refraction index dependent on wavelength?

The friction of electron in a media and the electronic field of a light (in certain wavelength(s)) changes the movement of the electron.

References

[1] List of laser types. https://en.wikipedia.org/wiki/List_of_laser_types

[2] Why do prisms work (why is refraction frequency dependent)? https://physics.stackexchange.com/questions/65812/why-do-prisms-work-why-is-refraction-frequency-dependent