A curveball should be thrown with similar arm speed as the fastball, but with a grip that produces top-spin on the ball so that the pitch moves sharply in a mostly downward direction.
How to hold a curveball:
A few different options for the curveball baseball pitching grip but still using the index and middle fingers and thumb. The middle finger is typically placed on the seam to the pinky side of the hand, with the index finger either on the ball or slightly off the ball or tucked underneath, depending on your grip of choice.
The action of the pitch is more defined by the wrist rather than the hand, though the hand is certainly important. The wrist supinates (or turns inward with the thumb pointed towards the head) at release and the middle finger rotates down while the thumb rotates up causing the top-spin.
Ball should start high on pitching arm side and should drop down and across the strike zone.
How to grip a curveball:
As with most pitches in baseball, pitching grips for curveballs vary. First understand what causes the ball to "curve." The pressure on a curveball is provided by the middle finger and thumb. The index finger is just along for the ride. Before release the hand should supinate (turn thumb in towards the head). At release, the middle finger rotates downward while the thumbs rotates upward, creating the top spin needed to make the pitch "curve" as it reaches the plate.
Let's start with probably the most standard of grips (pics are below). Place the middle finger on the seam on the right of the top horseshoe (vice versa for lefties). You can place the index finger on the ball but keep it loose.
Here are some pictures of different curveball grips...
Standard curveball grip
Another option, particularly a good one for youngsters, is the one-finger curveball (pics are below). The grip is identical to the standard grip except that the index finger is completely off the ball and in essence, pointing up. You don't have to tense up the index finger...it can be off the ball and just relaxed.
This grip is not used once the pitcher gets in high school because by that point, hitters are better at seeing the finger pointing up. When the hitter knows what's coming even before the ball is released, a pitcher has no chance. Seen below.
Curveball grip with index finger up
One more grip option is taking the index finger and bending it so that the knuckle or fingertip is on the ball (pics are below). This grip is much more advanced and is typically only used by older and more mature pitchers such as those in the latter years of high school, college and up.
This can lead to sharper and later break. The hardest thing to get used to with this grip is placing the fingertip on the ball. Nails need to short and manicured or you could end up with a nasty sore, infection, in-grown nail, blisters, etc.
Knuckle curveball grip
One last grip that I've seen is placing the ball so deep in the hand that there remains no space between the hand and the ball (pic is below). This grip is more rare than the others but some guys find it effective because there doesn't need to be a conscious effort to snap the ball with the thumb and middle finger. You just grip it and throw it.
No space curveball grip
How to pitch a curveball:
From hand break to right before release, think fastball and lead with the elbow. The forearm at release should be outside the elbow. The elbow should be at least as high as the shoulder if not a little higher. The hand should also be vertically in line with the forearm with the wrist straight, not turning the ball into the body causing the hand to "cup" the pitch.
A good visual when throwing a curveball is to picture the path of your hand after release to travel down and diagonal across your body. It is also the path that the ball should travel to the mitt. This will help emphasize the correct motion of your arm.
The pitch will most likely have some tilt as it travels to the target. This concept is best demonstrated by envisioning a clock. A true curveball is intended to travel from "12 to 6" on a clock, dropping straight down. I would say, though, that most pitchers have at least some degree of "tilt" on the pitch, causing the path of the pitch to move anywhere between the "12 to 6" or "1 to 7" or "2 to 8" range.
These baseball pitching grips for the curveball obviously vary and you should find one that suits your age and experience.
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