0
less than or equal to "t" less than or equal to
1
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what the heck? this is not even close!
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or so it looks. check again and you'll notice the bounds of the circle are the same as the bounds of the x axis. but why isn't it exactly equal? aren't we following the x axis?
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well, it is true that we're following the x axis, but we have to remember we are also following the y axis. this circle is defined on the xy plane, which means we have to find the coefficients of y as well.
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before i show you, try for a bit to see if you can get them yourself!
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if you tried, that's cool, if you didn't, that's a little less cool but still acceptable. we're still learning!
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0
less than or equal to "t" less than or equal to
1
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these are the coefficients. but what do we do now? we have them both separate, but we want one that covers the plane, and is actually 3 dimensional looking.
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luckily, this part is a lot easier than it seems. all we need to do is put the products together. remember that we've been multiplying these coefficients by either cos(2pi*t) or sin(2pi*t). this has to come with the coefficients.
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0
less than or equal to "t" less than or equal to
1
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now we know how to project any point on the (x,y) plane! if we wanted to project (0.3,0.2), we would just replace our c_x and c_y values.
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but what about 3d? what about the z axis?
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you're in luck! this part is actually easier than the (x,y) plane.
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notice what happens to this point as we drag our xy rotation.
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absolutely nothing! it stays fixed at zero, so we already have our first coefficient, zero.
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but what about the y value? well, it's only dependent on the xz rotation, and when that is zero, the point is at one. you know what that means!
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0
less than or equal to "t" less than or equal to
1
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our last coefficient is cos(r_xz). now we can write a formula for projection!
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don't worry if this looks complicated! it's just everything we've done compressed into one line that you can input coordinates into.
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what about functions though? well, that's the next section!
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functions
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behind the scenes
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Examples
Lines: Slope Intercept Form
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Lines: Point Slope Form
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Lines: Two Point Form
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Parabolas: Standard Form
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Parabolas: Vertex Form
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Parabolas: Standard Form + Tangent
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Trigonometry: Period and Amplitude
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Trigonometry: Phase
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Trigonometry: Wave Interference
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Trigonometry: Unit Circle
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Conic Sections: Circle
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Conic Sections: Parabola and Focus
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Conic Sections: Ellipse with Foci
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Conic Sections: Hyperbola
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Polar: Rose
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Polar: Logarithmic Spiral
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Polar: Limacon
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Polar: Conic Sections
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Parametric: Introduction
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Parametric: Cycloid
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Transformations: Translating a Function
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Transformations: Scaling a Function
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Transformations: Inverse of a Function
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Statistics: Linear Regression
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Statistics: Anscombe's Quartet
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Statistics: 4th Order Polynomial
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Lists: Family of sin Curves
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Lists: Curve Stitching
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Lists: Plotting a List of Points
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Calculus: Derivatives
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Calculus: Secant Line
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Calculus: Tangent Line
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Calculus: Taylor Expansion of sin(x)
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Calculus: Integrals
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Calculus: Integral with adjustable bounds
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Calculus: Fundamental Theorem of Calculus
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