Rotate the coordinate axes to remove the xy-term and sketch the graph of 3x+6√3xy+9y+(2√3 -4)x-(4√3 +2)y+20 . How can I solve it?

Lucid!

12:27 Same as last comment but remaining in radians until the last line # Exactly the same problem. Remain in radians throughout until we output the result in degrees # Prerequisites: GNU units, linux bash shell # For 1 degree on the GCD through Points (A,B) use arithmetic mean of the polar and equatorial radii # copy pasta: # SNIP------------------------------------------ phi_A_r=0.83077672 lambda_A_r=2.13506127 phi_B_r=0.37175513 lambda_B_r=2.75517676 # PART ONE: distance (A,B) = c or d in Gary's diagram a=`units -t "(pi/2 - $phi_B_r)"` b=`units -t "(pi/2 - $phi_A_r)"` gamma_r=`units -t "$lambda_B_r - $lambda_A_r"` cos_c=`units -t "cos ($a) cos ($b) + sin ($a) sin ($b) cos ($gamma_r)"` c=`units -t "acos($cos_c)" radian` C=`units -t "$c ((earthradius_equatorial + earthradius_polar) / 2)" nmile` C_statute_mile=`units -t "$C nmile" mile` printf "# %5.7f %5.7f %5.7f %5.7f " $phi_A_r $lambda_A_r $phi_B_r $lambda_B_r printf "# %-5.2f %-5.2f %-5.2f %5.7f %5.7f %5.2f %5.2f " $a $b $c_deg $gamma_r $cos_c $C $C_statute_mile # 0.8307767 2.1350613 0.3717551 2.7551768 # 1.20 0.74 38.78 0.6201155 0.7795150 2327.30 2678.21 # PART TWO: bearing (A,B) cos_alpha=`units -t "(cos ($a) - cos($c) cos ($b)) / (sin($c) sin ($b))"` alpha=`units -t "acos( $cos_alpha)" deg` bearing=`units -t "360 -$alpha"` printf "# %-5.2f %-5.2f %5.7f %5.2f %5.2f " $a $b $cos_alpha $alpha $bearing # SNIP------------------------------------------ # 1.20 0.74 38.78 0.6201155 0.7795150 2327.30 2678.21

12:27 distance then bearing # Remain in degrees throughout # Prerequisites: GNU units, linux bash shell # copy pasta: # SNIP------------------------------------------ phi_A=47.6 lambda_A=122.33 phi_B=21.3 lambda_B=157.86 # PART ONE: distance (A,B) = c or d in Gary's diagram a=`units -t "(90 - $phi_B)"` b=`units -t "(90 - $phi_A)"` gamma=`units -t "$lambda_B - $lambda_A"` cos_c=`units -t "cos ($a deg) cos ($b deg) + sin ($a deg) sin ($b deg) cos ($gamma deg)"` c=`units -t "acos($cos_c)" deg` C=`units -t "$c 60 nmile" nmile` C_statute_mile=`units -t "$c 60 nmile" mile` printf "# %5.2f %5.2f %5.2f %5.2f " $phi_A $lambda_A $phi_B $lambda_B printf "# %5.2f %5.2f %5.2f %5.7f %5.4f %5.2f %5.2f " $a $b $gamma $cos_c $c $C $C_statute_mile # 47.60 122.33 21.30 157.86 # 68.70 42.40 35.53 0.7795150 38.7838 2327.03 2677.90 # # Glen Gary: # 12:37 # PART TWO # Bearing (A,B) # Bearing for same trip, theta (A,B) wrt Grid North (North Pole = NP) # instead of theta, I use alpha to preserve notation symmetry # value d (aka c) is substituted in from above # cos_alpha=`units -t "(cos ($a deg) - cos($c deg) cos ($b deg)) / ( sin($c deg) sin ($b deg) )"` alpha=`units -t "acos( $cos_alpha)" deg` bearing=`units -t "360 -$alpha"` printf "# %5.2f %5.2f %5.7f %5.2f %5.2f " $a $b $cos_alpha $alpha $bearing # 68.70 42.40 -0.5028406 120.19 239.81 # SNIP------------------------------------------

6:31 Classical example of why it's horrible to teach math by memorization. You are using the angles in degrees! They have to be in radians and a believe the lengths of the sides have to be the lengths in a unit sphere! An attempt to derive this, would sort of show why this is true. I mean the circumference of a circle is 2*Pi*R and the rules sort of start to make sense.

Such good quality. Grateful.

Awsome video!

Excellent! Thanks for posting this. 👍

❤❤❤❤❤. Your lessons help me a lot. Thanks.

This really is the only video one needs to watch about rotation of axes. (2)

Unbelievable 😍, I understand 3 hrs lecture in 15 minutes. Thank you sir.

Idk why its so hard to find the formula on 3:51 on the internet. Thank you for helping me convert from and to rotated axis!

As you point out, the big picture really helps in setting all the details of celestial navigation into their context. Thank-you very much for your series on spherical trigonometry; very successful in achieving this goal!

Typically when choosing an Assumed Position a Navigator will NOT use the Dead Reckoning position. They will choose an AP on a whole degree of Latitude closest to the DR and a Longitude that will result in a whole degree of LHA(Local Hour Angle) from the GP(Geographical Position). This allows the Navigator to enter the Sight Reduction Tables with whole degrees, eliminating the need for interpolation… at least for that part of the calculation of the Navigational Triangle :)

I'm guessing all the algebra associated with x^2±x*y+y^2=1 is just the circle. It somehow exists because of the cosine law. So, I'm guessing there is an algebra for the ellipsoid to. Which is a rotated/skewed sphere from conics. Or maybe the cosine law only works in 2d. I would need to think about this.

thank you

Thank you for great videos, since you have much experience in this field, is there any new ways to represent Solids other then B rep and CSG and use winged edge data structure to represent them ? And could we preserve dependencies between edges, vertex and surface data on a different approach ?

This really is the only video one needs to watch about rotation of axes.

Hello, I found your video extremely helpful, as I am doing a small research work on spherical geometry. I was just wondering, what program/application did you use to create the visual models of spheres and figures generally speaking you used for this video?

GREATEST INTRO EVER SIR

Fascinating stuff, I am a geography student but I haven't studied the heavens before!

You sir are an ABSOLUTE LEGEND

As I find myself at the crossroads before setting off on the journey of writing my own CAD system, this video has been the single best resource I've ever come across To be honest, I had a sneaky suspicion that's how CAD programs worked under the hood, through experience with a lot of them, and some analytic geometry But this video really confirmed much about what I was suspecting and also clarified a few details such as why you need a different abstraction for a shell and a solid: because as you said, we need to be able to represent objects with internal cavities

Very interesting bit of trig. I appreciate you trying to go through one example of varying types of question that could be posed using these. Would GPS use any form of this math? If you get an AT2020USB+ or Blue Yeti Nano, your audio clarity would go way up.

That was beautiful, thank you.

Hello, thanks for making this video. What if my answer for the last question was (x'^2/4) + (y'^2/16) = 1. Would it be wrong? (Whether the answer is Yes or No, please state why)

I have been trying to figure these algorithms out on my own. I could not manage to find a book or get google to give me an original paper on the topic. I had managed to mostly figure out and find references for the 2D case and have that mostly working but I was really struggling with how to handle extrusion and 3D curves. Thank you so much for these videos, this was so hard to find!!! I didn't know curved surfaces used UV mappings, that makes so much sense now and it lets you decompose it all into 2 2D curves instead of all the broken mess I was attempting and failing.

These videos describing brep data structures and algorithms are very interesting. I was wondering if there were any practical source code examples you could point to for surface intersection and boolean operations?

What do you do if there are neither A or C terms?

I believe the rotation angle should be about 26.57 degrees and the sine of theta should be 1/(5)^.5

Hello everybody. I am not smart enough to understand the formular to rotate the ellipse, so i need more help. Can somebody add the missing part to this Basic listing please? SCREEN 12 COLOR 14 xmax = 640 ymax = 480 radx = 70 rady = 36 xm = xmax / 2 ym = ymax / 2 f = 2 pi = 3.141592654# FOR i = 1 TO (360 * f) c = (pi * i) / (180 * f) x = (COS(c) * radx) + xm y = (SIN(c) * rady) + ym PSET (x, y) NEXT i

Hi, great video. Is there a book that shows these techniques ?

Thanks sir The method u provided for rotation of axis is quite easy to understand as compared to the traditional method

Hi Glen Gray, Can you recommend a book on applied brep that describes all of its math. Which one can use to make a CAD software package 📦. Also please share the reference of this video. Best regards, FR

Good explanation! Just a little note: the right hand rule is tied to right-hand coordinate system that is not always the case, especially in a computer graphics (developers don't always know linear algebra and do questionable things). Math stays the same, but visual behavior may be confusing.

I actually can't get how we derive the angle of rotation's equation.

Excellent video. Thank you!

Nice song taste !!!

Dear sir How can rotate y=cos(x) With best wishes

Your voice is not clear there are disturbances and noise

outstanding !

Thank you so much! Finally I understand how the conversion between the systems of coordinates is done.

Thanks

I laughed at the song... That's a dad joke of the highest order... 50 points to gryffindor!

Thanks a lot

brep

I owe you an A-level

Thank you for your video, sir. However, I've got a problem. When we do the transformation to 3x^2 + 2xy+y^2 = 1, if I choose cot(2theta)=1-3/2=-1, I would got a ellipse, while I choose cot(2theta)= 3-1/2=1, I would not get the the xy item eliminated. It is really a headache. Would you please explain it. I will appreciate it.

wow what an amazing lecture on rotation/conics. covered everything i wanted. thanks sir.

Thank you (: