given three points, you can always find a circle passing through them all

1. draw lines from a to b to c
2. draw perpendicular bisectors (dashed)
3. the circle center is their intersection point
4. the circle radius is the distance from the center to any of the three points
this works because

• the circle center is by definition, the same distance to all three points
• given a pair of points, the region where the distance to both is the same, is a perpendicular line between them
• the intersection point is therefore where all three are the same
note that if all three points lie along a line, the circle is undefined! the bisectors would never intersect, so there is no circle passing through all points

(also if you comment about this because you don't read threads I will link you this tweet and make you feel *shame*)
if we interpret the three points as vertices of a triangle, this circle is usually called the circumcircle

the category of polygons where all points lie in a circle are called cyclic polygons! and so, by definition, all triangles are cyclic polygons
like comment and inscribe for more circular facts

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More from @FreyaHolmer

2 Sep
there are lot of misconceptions on high refresh rate monitors, so, quick facts!

• the eye doesn't have a "max framerate"
• the eye can absolutely distinguish between 144Hz and 1440Hz monitors

the eye is continuously picking up light - if something is moving really fast, then it won't just get a "frame", it will smear in a blur, the size of which depends on the lighting conditions and the properties of your own eye

so - what if something is moving fast on a monitor?
try moving your cursor in a large circle quickly

do you see gaps between the cursors?

double your refresh rate and you would get one more cursor in the middle of those gaps

quadruple and you get three cursors covering that gap

see where this is going?
Read 7 tweets
29 Mar
barycentric hecking interpolation!!

a little thread on a neat way to interpolate across a triangle!

say you have three points, each with a color - how do you get the blended color for an arbitrary point inside? 🤔
byerp (coining this term now~) uses the area of the triangle on the *opposite* side of each vertex, as a fraction of the total area, to determine how much influence each vertex has!
these three influences, or, weights, together form the barycentric coordinate of that point
we can then multiply each weight with their corresponding color, and add them all up, to get the final blended color at that coordinate!

since models in games are made of triangles, this is actually how mesh data like normals, UVs, vertex colors etc. is interpolated by your GPU!
Read 5 tweets
27 Feb
my favorite way to see if a point is inside or outside a path, is using its winding number🍥

traverse the path from the perspective of a point and add up the amount of turning along the way

if it made a full turn, it's inside
if it wound back to 0, it's outside

it's so neat~💖
this is one of my favorite techniques, because it works not only for convex paths, but for concave ones too, which, is kinda rare and beautiful

plus it's real fast if you measure quadrant traversal rather than real angles~

more info here…
if you want more math and game dev shenanigans, do consider supporting me on patreon, if you are able to and want to 💖
Read 5 tweets
21 Dec 19
what just happened to JK Rowling?

the definitive thread on questions like

• why is everyone upset?
• what’s wrong with saying sex is real?
• she seems supportive, how is her tweet bad? well as why we're now talking about Hatsune Miku

so, first off, here's her tweet!
it starts off seemingly supportive of LGBT rights, and cryptically ends referring to someone who was forced out of their job for "stating sex is real", followed by a #IStandWithMaya hashtag

so, who is Maya and what happened?
this is my source, by the way, thought I'd link it here if you want to read it yourself

Maya Forstater v CGD Judgement 2019…

so, let's jump in and see what this is about
Read 24 tweets
23 Sep 19
A quick guide to Lerp~✨

Lerp is used to blend between two things!

It has three inputs: ( a, b, t )
a = start value
b = end value
t = how far we should go from start to end, as a percentage

Here it is in action!⚡blending a value, position, color and an angle!
If you're using Unity - make sure you use the Lerp appropriate for your use-case!

Mahtf.Lerp - values
Mathf.LerpAngle - angles
Color.Lerp - colors
Vector#.Lerp - positions
Vector#.Slerp - directions
Quaternion.Slerp - rotations

here's how to lerp mathematically!

a(1-t) + bt
weighted sum interpretation, where both contribute a total of 1, trading off influence

a + t(b-a)
this one originates from a, moving toward b by adding the displacement scaled by t

oh, and, these two are mathematically equivalent!
Read 7 tweets
12 Sep 19
found a sphere at the office!🔴

here's a quick demo of metallic reflections~

note how the color of the window's reflection on the sphere changes
even though the sphere is metallic red, it doesn't exclusively reflect red and absorb the rest, it depends on the angle

at a high grazing angle - only reds reflect

at a low grazing angle - all wavelengths reflect, hence all colors in the light from the window is preserved!
by the way - this happens with transparent materials as well, they too reflect light at low grazing angles!

for instance, look at the difference in transparency of the water at the shore vs the perfect mirror at the much lower angles further away

📸 by @nicolesy
Read 5 tweets

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