Anime and cartoons are only different forms of animation
Despite the title, animation is not automatically anime. Both cartoons and anime utilize animation to create them. The only true difference between them is the art style used.
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Media like Disney's Frozen or games like Overwatch are usually what people think about when people talk about animation. Characters rendered with computers and 3d imagery. Other times they think about anime exclusively but that's not what animation is, and it's not so exclusive as to only pertain to anime. Animation is an arrangement of photos put together that give the image or illusion of movement. With modern technology, many examples of animation are made with computer programs but the core idea is that the images look like their moving as if they were real. With animation, the technical part behind it is the flow of the body, the movement of the characters and how fluid those movements are. Many people confuse this with the art style different animators use. How the artwork of a show or movie looks is the style while animation focuses solely on the motion of the arts characters. Not liking the art style does not mean that the animation is poor. Both cartoons and anime are animations that typically use different styles of art which is part of the reason why they're usually put into separate categories. However, the core pieces of what make animation, 'animation' are all there. They utilize music, voices, sounds, color, motion, and lighting, all in their creation to make something. The sound might be different, designs varying, and the fluidity of the characters not being quite smooth. They create something though, that brings joy to the people who watch it.
Both anime and cartoons are considered forms of animation but they are also considered subcategories. Cartoons and anime have their own unique characteristics which sets them apart from other type of animation. It's the same idea between the difference of 2D animation and CGI animation. With cartoons, the images that make up the animation are almost always 2D. Most people think that cartoons only mean childish shows that are targeted towards children like SpongeBob SquarePants or Dora the Explorer. That's not the case though. Cartoons were originally meant for humor, appearing as illustrations as inflamatory propaganda or comic strips like the Peanuts. People can see this in trend has continued with popular shows that fall into the cartoon category. Shows like Family Guy, Futurama, and the Simpsons. They aren't always what people would consider child appropriate cartoons but not all cartoons are meant for that. With anime on the other hand, the style people usually see is bigger eyes, intense fight scenes, and an aesthetic that usually falls into 'cool' or 'cute'. This is for anime with a younger target audience population though. Shows with tropes like 'magical girl' or 'the power of friendship will defeat our enemies.' In darker animes though, ones like Death Note, Tokyo Ghoul, and Attack on Titan and considerably tame for some of the dark and emotional animes that exist. These two categories exist because despite the fact they are similar and take many different forms, at their core there's a different focus. Cartoons all contain a comedic aspect to them, whether it's mild humor or dark humor. It's supposed to make people smile or laugh. With anime on the other hand, the focus is more emotional.
[P1] While different art styles are utilized for different shows they are still forms of animation [P2] Art giving the illusion of movement is what makes up animation
Rejecting the premises
[RP] Animation and cartoons have different subcategories