Chaitanya's Random Pages

February 28, 2015

Large US+Canada box office openings

Filed under: movies and TV — ckrao @ 1:38 pm

The site boxofficemojo.com lists the opening weekend grosses of movies in the US and Canada dating back to the early 1980s. Via this page on top opening weekends, I worked out movies that at their time of release attained the n’th highest grossing opening weekend where n ranges from 1 to 10 (all dollar amounts in $US). It gives a perspective on how big some movies were at the time. It also shows how movie grosses have grown through inflation and more frontloading over the years. Note that only opening weekends are shown here – for example Superman’s third weekend was once the largest grossing weekend at the time, but is not listed here.

 

n=1 (i.e. current and previous record-breaking openings):

Title Opening Date (mm/dd/yyyy)
Marvel’s The Avengers $207,438,708 5/4/2012
Harry Potter and the Deathly Hallows Part 2 $169,189,427 7/15/2011
The Dark Knight $158,411,483 7/18/2008
Spider-Man 3 $151,116,516 5/4/2007
Pirates of the Caribbean: Dead Man’s Chest $135,634,554 7/7/2006
Spider-Man $114,844,116 5/3/2002
Harry Potter and the Sorcerer’s Stone $90,294,621 11/16/2001
The Lost World: Jurassic Park $72,132,785 5/23/1997
Batman Forever $52,784,433 6/16/1995
Jurassic Park $47,026,828 6/11/1993
Batman Returns $45,687,711 6/19/1992
Batman $40,489,746 6/23/1989
Ghostbusters II $29,472,894 6/16/1989
Indiana Jones and the Last Crusade $29,355,021 5/24/1989
Beverly Hills Cop II $26,348,555 5/20/1987
Indiana Jones and the Temple of Doom $25,337,110 5/23/1984
Return of the Jedi $23,019,618 5/25/1983
Star Trek II: The Wrath of Khan $14,347,221 6/4/1982
Superman II $14,100,523 6/19/1981
Star Trek: The Motion Picture $11,926,421 12/7/1979
Every Which Way But Loose $10,272,294 12/20/1978

 

n=2 (i.e. the second largest opening at the time)

Title Opening Date
Iron Man 3 $174,144,585 5/3/2013
Star Wars: Episode III – Revenge of the Sith $108,435,841 5/19/2005
Shrek 2 $108,037,878 5/19/2004
The Matrix Reloaded $91,774,413 5/15/2003
Planet of the Apes (2001) $68,532,960 7/27/2001
The Mummy Returns $68,139,035 5/4/2001
Star Wars: Episode I – The Phantom Menace $64,820,970 5/19/1999
Independence Day $50,228,264 7/3/1996
Lethal Weapon 3 $33,243,086 5/15/1992
Terminator 2: Judgment Day $31,765,506 7/3/1991
Rocky III $12,431,486 5/28/1982
The Cannonball Run $11,765,654 6/19/1981
Smokey and the Bandit II $10,883,835 8/15/1980
The Empire Strikes Back $10,840,307 6/20/1980

 

n=3:

Title Opening Date
The Dark Knight Rises $160,887,295 7/20/2012
The Hunger Games $152,535,747 3/23/2012
The Twilight Saga: New Moon $142,839,137 11/20/2009
Shrek the Third $121,629,270 5/18/2007
Harry Potter and the Prisoner of Azkaban $93,687,367 6/4/2004
Harry Potter and the Chamber of Secrets $88,357,488 11/15/2002
Star Wars: Episode II – Attack of the Clones $80,027,814 5/16/2002
Hannibal $58,003,121 2/9/2001
Mission: Impossible II $57,845,297 5/24/2000
Toy Story 2 $57,388,839 11/24/1999
Austin Powers: The Spy Who Shagged Me $54,917,604 6/11/1999
Men in Black $51,068,455 7/2/1997
The Lion King $40,888,194 6/24/1994
Rambo: First Blood Part II $20,176,217 5/22/1985
Star Trek III: The Search for Spock $16,673,295 6/1/1984

 

n=4:

Title Opening Date
X-Men: The Last Stand $102,750,665 5/26/2006
Harry Potter and the Goblet of Fire $102,685,961 11/18/2005
X2: X-Men United $85,558,731 5/2/2003
Austin Powers in Goldmember $73,071,188 7/26/2002
Rush Hour 2 $67,408,222 8/3/2001
Pearl Harbor $59,078,912 5/25/2001
Mission: Impossible $45,436,830 5/22/1996
Twister $41,059,405 5/10/1996
Back to the Future Part II $27,835,125 11/22/1989
Rocky IV $19,991,537 11/27/1985
Beverly Hills Cop $15,214,805 12/5/1984
Jaws 3-D $13,422,500 7/22/1983
Superman III $13,352,357 6/17/1983

 

n=5:

Title Opening Date
The Twilight Saga: Breaking Dawn Part 1 $138,122,261 11/18/2011
Iron Man 2 $128,122,480 5/7/2010
Pirates of the Caribbean: At World’s End $114,732,820 5/25/2007
How the Grinch Stole Christmas $55,082,330 11/17/2000
Interview with the Vampire $36,389,705 11/11/1994
Home Alone 2: Lost in New York $31,126,882 11/20/1992
Bram Stoker’s Dracula $30,521,679 11/13/1992
Star Trek IV: The Voyage Home $16,881,888 11/26/1986
The Best Little Whorehouse in Texas $11,874,268 7/23/1982
E.T.: The Extra-Terrestrial $11,835,389 6/11/1982

 

n=6:

Title Opening Date
The Hunger Games: Catching Fire $158,074,286 11/22/2013
Harry Potter and the Deathly Hallows Part 1 $125,017,372 11/19/2010
Alice in Wonderland (2010) $116,101,023 3/5/2010
The Passion of the Christ $83,848,082 2/25/2004
Monsters, Inc. $62,577,067 11/2/2001
X-Men $54,471,475 7/14/2000
Ace Ventura: When Nature Calls $37,804,076 11/10/1995
Robin Hood: Prince of Thieves $25,625,602 6/14/1991
Total Recall $25,533,700 6/1/1990
Teenage Mutant Ninja Turtles $25,398,367 3/30/1990
Ghostbusters $13,578,151 6/8/1984
Staying Alive $12,146,143 7/15/1983

 

n=7:

Title Opening Date
Transformers: Revenge of the Fallen $108,966,307 6/24/2009
Spider-Man 2 $88,156,227 6/30/2004
Batman and Robin $42,872,605 6/20/1997
Lethal Weapon 2 $20,388,800 7/7/1989
Star Trek V: The Final Frontier $17,375,648 6/9/1989

 

n=8:

Title Opening Date
The Twilight Saga: Breaking Dawn Part 2 $141,067,634 11/16/2012
The Lord of the Rings: The Return of the King $72,629,713 12/17/2003
Godzilla $44,047,541 5/20/1998
The Flintstones $29,688,730 5/27/1994
Gremlins $12,511,634 6/8/1984

 

n=9:

Title Opening Date
Finding Nemo $70,251,710 5/30/2003
The Mummy $43,369,635 5/7/1999
Deep Impact $41,152,375 5/8/1998

 

n=10:

Title Opening Date
Toy Story 3 $110,307,189 6/18/2010
Indiana Jones and the Kingdom of the Crystal Skull $100,137,835 5/22/2008
Iron Man $98,618,668 5/2/2008
Dick Tracy $22,543,911 6/15/1990

(To create the above lists the movie lists in decreasing order of gross were pasted into Excel and the opening weekend date was converted to a number by creating a new column with formula =–TEXT(,”mm/dd/yyyy”). This was then converted to a rank by a countif formula to count the number of occurrences with higher gross that predated each movie. Finally a filter was applied to select ranks 1 to 10.)

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February 27, 2015

Cross sections of a cube

Filed under: mathematics — ckrao @ 9:59 pm
When a plane intersects a cube there is a variety of shapes of the resulting cross section.
  • a single point (a vertex of the cube)
  • a line segment (an edge of the cube)
  • a triangle (if three adjacent faces of the cube are intersected)
  • a parallelogram (if two pairs of opposite faces are intersected – this includes a rhombus or rectangle)
  • a trapezium (if two pairs of
  • a pentagon (if the plane meets all but one face of the cube)
  • a hexagon (if the plane meets all faces of the cube)

The last five of these (the non-degenerate cases) are illustrated below and at http://cococubed.asu.edu/images/raybox/five_shapes_800.png . Some are demonstrated in the video below too.

cs1 cs2 cs3 cs4 cs5

One can use [1] to experiment interactively with cross sections given points on the edges or faces, while [2] shows how to complete the cross section geometrically if one is given three points on the edges.

Let us be systematic in determining properties of the cross sections above. Firstly, if the plane is parallel to an edge (any of four parallel edges), the cross section can be seen to be a line or rectangle with the longer dimension of length at most \sqrt{2} times the other. That rectangle becomes a square if the plane is parallel to a face.

If the plane is not parallel to a face, we may set up a coordinate system where a unit cube is placed in the first octant aligned with the coordinate axes and the normal to the plane has positive x, y and z coordinates. In other words, we may assume the plane has equation ax + by + cz = 1 and intercepts at (1/a,0,0), (0,1/b,0) and (0,0,1/c), where a, b, c are positive.

coordsystem

The cross section satisfies ax + by + cz = 1 and the inequalities 0 \leq x \leq 1, 0 \leq y \leq 1 and 0 \leq z \leq 1. This can be considered the intersection of the two regions

\displaystyle ax + by + cz = 1, 0 \leq x, 0 \leq y, 0 \leq z,

\displaystyle ax + by + cz = 1, x \leq 1, y \leq 1, z \leq 1,

each of which is an acute-angled triangle in the same plane (acute because one can show that the sum of the squares of any two sides is strictly greater than the square of the third side). Note that the triangles have parallel corresponding sides, being bounded by the pairs of parallel faces of the cube x = 0, x=1, y = 0, y= 1, z=0, z=1. Hence the two triangles are oppositely similar with a centre of similarity.

The following diagram shows the coordinates of the vertices of the two triangles, which in this case intersect in a hexagon.

2triangles

The centre of similarity of the two triangles is the intersection of two lines joining corresponding sides – this can be found to be the point (1/(a+b+c), 1/(a+b+c), 1/(a+b+c)), which is the intersection of the unit cube’s diagonal from the origin (to (1,1,1)) and the plane ax + by + cz = 1.

Side lengths of the triangles and distances between corresponding parallel sides may be found by Pythagoras’ theorem and are shown below for one pair of corresponding sides (the remaining lengths can be found by cyclically permuting a,b,c).

lengthsofsides

To sum up, all of the possible cross sections of a cube where the plane is not parallel to an edge can be described by the intersection of two oppositely similar triangles with corresponding sides parallel.

The type of polygon obtained depends on which vertices of the figure below are selected, as determined by the values of a,b,c.constraints

In this figure a vertex for the cross-sectional polygon is chosen if the constraint associated with it is satisfied. A red vertex has a conflicting constraint with its neighbouring two blue vertices, so either a red point or one or more blue points in this area can be chosen. Note that for the plane to intersect the cube at all we require (1,1,1) to be on the different side of the plane from the origin, or in other words, a + b + c \geq 1.

Let us look at a few examples. Firstly, if a, b, c are all greater than 1 we choose the following triangle.

tricase

Similarly if a+b, b+c, c+a all are less than 1, the oppositely similar triangle on the red vertices would be chosen.

Next, if c > 1, a < 1, a+b < 1 we obtain the following parallelogram.

parallelogramcase

 

 

If c > 1, a < 1, a+b > 1 we obtain either a pentagon (parallelogram truncated at a vertex) or a trapezium depending on whether b < 1 or b \geq 1 respectively.

b < 1:

pentagoncaseb\geq 1:
trapeziumcase

Finally, if a, b, c are less than 1 and a+b, b+c, c+a are greater than 1, we obtain a hexagon.

hexagoncase

For details on calculating the areas of such polygons refer to [3], especially the method applying the area cosine principle that relates an area of a figure to its projection. For calculating volumes related to regions obtained by the cross section refer to [4].

References

[1] Cross Sections of a Cube: http://www.wou.edu/~burtonl/flash/sandbox.html

[2] Episode 16 – Cross sections of a cube: http://sectioneurosens.free.fr/docs/premiere/s02e16s.pdf

[3] calculus – Area of the polygon formed by cutting a cube with a plane – Mathematics Stack Exchange

[4] integration – Volume of cube section above intersection with plane – Mathematics Stack Exchange

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