Parametric curves | Multivariable calculus | Khan Academy

When a function has a one-dimensional input, but a multidimensional output, you can think of it as drawing a curve in space.

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[lavon]

Wait is this Grant Sanderson (3blue1brown)? That's awesome! LOL

[MrMLehman]

Grant, thanks for making these Multivariable Calculus videos. They're terrific.

[johnstillman]

really incredible skill with graphing but the volume keeps changing levels video to video

[singalo]

Khan academy, will you soon start teaching abstract/pure mathematics? Its hard to find easy resources online for these subjects

[berryzhang]

If grant was my professor I’d never skip lecture!

[sgut]

At 5:43, shouldn't t=1 still be in the first quadrant? After the origin, the first time it crosses the y=axis is at t=pi/2, isn't it?

[swkit]

I was using t= 2(n)/pi that generate exact same graph instead of using 0<t<10

[riyazhudda]

Grant, thank you. What I am even more interested in is how do you draw these cool graphics.

[umeshgj]

You made the world a better place to live in...this might lead to find a new world to live in....

[sandeepm]

Awesome method of teaching. Terrific work.

[thhar]

Calling them "pi-halves" is so cute!

[sorryall]

The plot of t*cos(t), t*sin(t) is equivalent as if you drew a circle with a compass whose legs are being constantly stretched.

[koushikreddy]

What is the function that draws the same graph??

[GundoganFatih]

Which tool are you using to draw these parametric functions?

[jiuzhe-xz]

parameter is just kind of a fancy word for input

[rxtez]

what software do you use to plot these? Excel or something else?

[manassehjustice]

pls someone should help me out with this assignment
Calculate the frenet apparatus for the parameterized curves alpha(t) = (3t-t³, 3t², 3t+t³)

[quosswimblik]

There's a type of parametric curve everyone has missed. graph a slice of a circle no bigger than a semicircle then change one of the radius lengths. Now progress the one radius length to the other radius length in the simplest linear way you should find that for any triangle every side has a corresponding curve. In 3D this also works for curved surfaces on the faces of any tetrahedron. Not only that but you can change how the one radius progresses to the other radius and this even works in 3D if you want to say add bumpiness or roughness to a human face or want to flatten the curvature a bit to represent well the face of wall. Not only this but the surface setting on the triangle can also represent coloring of the surface. In terms of video compression this technique can allow for a very upscalable image be the curvature of smaller detail not as advanced you could also save a lot on data size of the video stream.

[oscarC]

Can this explain the shape of galaxies?

[zes]

wrg, no problx about that, can do anyx